Let's agree first that a site request is ONLY a request and not a guarantee. It has to be available and empty the day you arrive in order for you to be assigned to it (or the guest currently on it is leaving the same day you arrive).
When you make a reservation over the phone, you can certainly ask the CM to "add a note" to request a specific site or loop number. Your reservation is ONLY for a certain level of site (Tent, Full, Preferred, Premium, Premium Meadows). There is not a way to make a request when making a reservation with the on-line booking tool. In those cases you have to call back some time, any time, before arrival and ask to have the note added. The computer system pre-assigns your reservation a site number in the days before you arrive based on what sites should be vacant for your level. However, the notes/requests are passed down to the Fort as well as site numbers and a CM reviews that and tries to make as many requests come true with manual changes before arrival based on availability (the CM's really do try to deliver the magic in this regard).
I'm a person that always requests a specific loop in my reservation and then when I arrive I try to ask about 2-3 specific sites in that loop. If you just ask for site #1624, for example, and it's not available the CM has no idea why you wanted it and has less ability to make the magic happen. Maybe it's very close to the Comfort Station (your intent) and #1625 might be available. Maybe you wanted to be in 1600 loop. But they would see that #1624 is unavailable and put you in another Full loop.
I request a loop and then try to check in as early as possible on the arrival day. I admit I take it to the extreme. The "day" at FW starts at 6am and I've walked in before 7am for about the last 8 trips. Nobody's there, the CM just came on duty and is fresh, and a conversation (negotiation?) is possible. They would see my request for loop 100, for example. And I offer up in conversation, "gee it would be great if #114, #116, or #127 is available". The CM would say that it might be possible but I'd have to wait till the guest left that morning. Or they might say, none of those are available but I do have #101 available. That's what happened to me earlier this month. Made a Tent reservation, requested 1500 loop, got there at 630am on arrival day and asked for 1511 (best site in the loop IMO). For the first time, in four tries, I got it - had to wait though for the guest to leave and site to be prepped. A site was ready in 2000 early that morning but I passed on it.
The reason I suggest arriving as early as possible is the later you arrive, the fewer your options (if you care). 1500 is a popular (and small) loop with only 21 sites. If five sites are available for assignment on arrival day and six people requested 1500, somebody's not going to get it (the last person to arrive - I've been that guy and that's when I learned my lesson). I got bumped to 2000 loop and that was the last time I had to stay in it. I make no plans for parks on arrival day and only dinner ADR's by which time I should have gotten whatever I'm gonna get. If you're a guest who is happy to get whatever you get, no problem. But it won't happen unless you ask and the earlier you arrive, the more likely that it can happen.
Sorry to be so wordy but as long as you check in early and have a couple options for sites, the chances are good that you'll get something you're happy with.
Bama Ed
PS - weekends Jan-May and Oct-Dec book up pretty early with Florida locals making their 2-3 day plans especially in the Full and Preferred levels of sites. So book early if you plan to travel then. What happens if you want a week, it will not be available because Sun-Thurs are there but Fri and/or Sat are not.