Now, I last did a passport in 2009, so some of these might be slightly different. Your best bet is to check out the Facts page on USEF. You'll need access to the USEF webpage throughout the process, so if your computer isn't handling it, you need to find one that does. Try changing your browser perhaps?
Anyways, when I did it, you first had to register the horse with a Lifetime Recording Number. Horse and Rider will also need USEF membership and High Performance numbers. Then fill out all the paperwork and send it in to USEF to request a passport. It can be 4-6 weeks before they process and send back a blank passport. (For instance, I sent in a Passport Revalidation form in late January. This form is very simple, and they just reissue a sticker. They received it last week but it's in the queue to be processed late next week...)
Once you get the blank passport, call your vet and get him to fill out the passport. Make sure he is familiar with the process, as the USEF frowns upon mistakes in it. If he's not, I highly recommend working with a vet who IS familiar. Make sure your horse also gets the initial shot and booster of the flu vaccine and that your vet records these in the passport. There's a specified number of days required between, but I can't remember how many.
Send the completed passport BACK to the USEF. Again, it can take 4-6 weeks to process, perhaps longer if they are busy (and this is a busy time of year, since everyone is thinking ahead to their first FEI). Only then will they send it back to you, and you must have it in hand at your first in-barns.
Make SURE that you get the booster every six months and update your passport accordingly. Also make sure that the booster is not given ten days before you hit the show grounds, or mistakenly recorded as such. I have seen some major, major issues over this (including eliminations after dressage when the snafu was caught, calls to Sweden begging for leniency, tearful kids and parents, etc.).
Ditto what Divine said about starting early. It is possible to get a passport quickly if need be (we once got one in 9 days from start to finish), but this is a costly and nerve wrecking way to go. Do it while you still have plenty of time.
Also, as Divine noted, mistakes are frowned upon and may result in the approval of your passport being delayed. To avoid this, make copies of the pages your vet needs to complete while they are still blank, have your vet them out as if they were the actual pages in the passport, and then fax them to the USEF for their review. Once they approve them, have your vet transfer the information from the copies into the passport. This may add some time to the process, but it will help insure that the final approval of your passport is not delayed due to mistakes. This is especially helpful if you are working with a vet who is not familiar with the process.
Two other important things to remember. First, use block letters (all caps) when entering information into the passport. Make sure your vet knows to do this as well. Second, be sure to use the correctly colored pen when completing the outline diagrams. Unless things have changed, markings must be drawn in red with whorls, scars, and other distinguishing features drawn in black.
If the Number 2 pencil is so popular, why is it still number 2?