I had to request six transcripts. For three of them, I filled out and faxed the school's transcript request form, and added a line on it requesting that they include the enclosed AMCAS form (which I included in my fax) to be sent with a copy of my transcript to AMCAS. My transcripts for those three schools were received by AMCAS ASAP (I think they only took a week). Two other schools would only take transcript requests in person, or by mail with a check. Considering that these are schools located all over the country, I obviously opted for the mail route. I printed out and enclosed each school's personal transcript request form, and again put in a note requesting that they include the enclosed AMCAS transcript form when they send the transcript to AMCAS. Both of those transcripts have also been received by AMCAS now, but it took MUCH longer (almost 3 weeks from date of request), probably due to the schools cashing my check and snail mail.
My sixth transcript I got lazy on. I simply used the school's online request page, and included my AMCAS ID number at the end of the "To:" line for AMCAS' address. I did not include the AMCAS form. That transcript was received and logged by AMCAS ASAP. I think it only took one week from online request to AMCAS email notification to me that they received it.
If you can fax a transcript request to your school, just ask that they enclose the AMCAS form with your transcript, and attach it. I did it without even asking any of my schools, and didn't run into any problems. I'm sure your school has seen such a request before. If you're worried about it, call the registrar and ask them if they'd do it. Alternatively, you can do what I did for school number 6. Just request the transcript without the form (but try in some way to get your AMCAS ID on there), and if it hasn't been received by AMCAS within two weeks, do it by fax or mail. I think you should go with a fax. The only reason I had the last school's requested online was that it would be a lot faster than snail mail, and I wanted all my transcripts in before June 5th.
Good luck!