Thanks for your inquiry. Trying to contact the National Archives will, as you say, take a long time to get an answer, if only thanks to covid-19. (See https://www.archives.gov/coronavirus for information from the National Archives per covid.) Before trying that, here are some things you can do online.
It sounds like you know the names of the three ships in which your grandfather served. If so, try this website to get basic information about the construction and some history of each ship: www.shipbuilding.com. There is a search button at the very bottom of the page. You should get a record of when and where the ship was constructed (name of shipyard, date of keel-laying, date of launch, date of completion) and at least some indication of the ship's eventual fate.
For more detail about each ship, go to http://www.mariners-l.co.uk/LibIndex.html and search through the table for the name of each ship. Note that the table is arrayed by the FIRST name of the ship, a traditional way of alphabetizing names of merchant ships. (Therefore, SS JOHN SMITH will be found under John, not under Smith.) There may be a little more information about each ship, particularly if the ship had a postwar career. There will likely be more information about the ship's eventual fate than noted at shipbuilding.com.
Then go to this excellent website, http://www.convoyweb.org.uk/hague/index.html, and perform a Ship Search for the name of each ship. From this you will get a list of all the wartime convoys in which the ship sailed, the originating and destination port, the inclusive dates of the convoy from departure to arrival, and sometime a very little bit of information about the ship itself, such as the cargo carried or its position within the convoy. You apparently have an idea of the dates your grandfather was aboard each ship so by cross-referencing those dates against the dates of the convoy(s) in which the ship sailed, you can get an idea of where he was at any given time (if only "at sea").
At the bottom of the results page for each ship there will be a phrase, "To continue the search for OTHER voyages of [name of ship], click here." When you click there you will get a complete listing of the ship's whereabouts during the war, in chronological order, including instances in which the ship sailed independently, i.e., not in convoy and without warship escort. That list is even more useful than the list by convoys.
Be aware that while the records found in ConvoyWeb are very good for ships operating in the Atlantic and Mediterranean and to a lesser extent in the Caribbean, the site is seriously lacking on ship movements in the Pacific. If any of your grandfather's ships operated in the Pacific, ConvoyWeb may have nothing.
There is a similar website, http://warsailors.com/search.html, which in general is not nearly as comprehensive as ConvoyWeb but on occasion has much more information about an individual ship than that found in ConvoyWeb.
Here's another idea which may involve some out-of-pocket costs. Get a short-term subscription to ancestry.com. This website is used extensively for genealogical research but I have found that it contains databases of the names of passengers and, more importantly, crew members of ships arriving at certain U.S. ports of entry following a foreign voyage. The records of the port of New York are especially extensive.
Make your search under Immigration and Travel (https://www.ancestry.com/search/categories/40/), enter your grandfather's name and specify an Arrival date for the year 1943 plus or minus two years (thereby covering all World War II arrivals). If you find a record it will show the name of the ship he was aboard, the date the ship arrived in a U.S. port, and the names of his shipmates, among other things. If your grandfather's name was fairly unique your chances of finding him are better than if he was named John Green but also be aware misspellings are common in these records. Good stuff.
Ancestry.com often has introductory sales in which you can establish an account for a short time (two weeks, maybe), perform your searches and discontinue your account before the introductory period ends, thereby getting away without paying a cent. Otherwise it looks like you can get a full-price subscription right now for $25/month. You can decide whether it's worth the cost. Or, send me his full name and the names of his ships and I'll see what I can find for you, as I have a subscription.
OK, back to your original question. You should contact the National Archives at:
National Archives Textual Reference Branch (RDT2)
8601 Adelphi Road
College Park, MD 20740-6001
What you are seeking specifically are known as Armed Guard logs (sometimes referred to as Armed Guard commanding officers' reports). These logs are found in the National Archives Record Group 24: Records of the Bureau of Naval Personnel. FYI, these are "... daily logbooks and reports relating to routine inspections, personnel, and communications aboard naval vessels. These logs were prepared under the direction of the armed guard commander aboard each ship, and comprise a brief daily account of events of the armed guard crew including mustering, disciplinary actions, and security matters." See https://catalog.archives.gov/id/6434900. At the bottom of this web page you can see the list of 741 "containers" (boxes) of Armed Guard logs held by the Archives, arranged by name of ship. You will have to specify the name of the ship and, ideally, the date a particular voyage ended (as found in ConvoyWeb) or at least a range of dates of interest ("all voyages ending in 1944" for example). The last day your grandfather was assigned to a ship will correspond closely, within a day or two, of the end of a voyage. If he was on multiple voyages with a given ship, which was common, you will need to know the date of the end of each voyage, or supply a date range.
At the conclusion of each voyage the commanding officer of each Armed Guard detachment aboard a merchant ship was obliged to submit to the Chief of Naval Operations, through the Port Director of the port in which a voyage ended, a report detailing the daily activities of the Armed Guard unit during that voyage. At the minimum there will be a list of the names of all the Armed Guard sailors and officers in the detachment, plus day-by-day activities (many very routine and unremarkable), including the times, if any, the ship was at battle stations or under attack. The length, quality and detail found in these reports vary widely depending on the diligence of the commanding officer but often contain real gems of information. The Armed Guard website has a selection of Armed Guard officers' reports beginning at http://www.armed-guard.com/reports.html.
Assuming the Archives is able to process a request for information from you, there will be fees for research time, photocopying and mailing. The Archives staff will inform you of the likely cost before beginning work on your request. If you are within driving distance of College Park, MD, a suburb of Washington, DC, you can perform your own research in person without charge (other than photocopying), provided the facility is open to the public per covid restrictions.
I hope this is enough to get you started. Let me know if you have additional questions or if I can help in other ways.
Ron Carlson, Webmaster
Armed Guard / Merchant Marine website