Vatican & St. Peters questions:
Is there a shuttle that runs from St. Peters to the entrance of the Vatican?
Is there an elevator in the Vatican? (my husband has a hard time walking up stairs).
Is 8 AM a good time to visit the Vatican or
will the lines be shorter around 1 PM

Your question is a bit confusing. The St. Peter's Basilica is in the Vatican State. Are you asking about shuttle service to the Vatican Museums from St. Peter's Square? What month are you going? Opening times vary per season and so do crowds.

The entrance to the Vatican museum is about a 15 minute walk from St Peter's Sq.

I don't remember a shuttle. You can take a taxi but depending upon your luck there may be a line going back almost to the Square.

I think the Vatican museum open around 9AM give or take 15 minutes.

I never took it but there once was a Vatican bus that did go between St. Peter's and the museum. From what I read and recall it went thru the Vatican Gardens to a different museum entrance allowing you to bypass the long outside line.
I asked about it at the info booth a couple years ago but was told it was discontinued.
I had better luck going later in the day on my 3rd visit.
(If nothing has changed!) And you do the museum 1st, end your visit at the Sistine Chapel. In the far right corner of the Chapel is an exit marked "For Tour Groups Only". Just follow a group out (honestly it's no big deal and no one really cares and this will put you outside alongside St. Peter's Basilica. It will save you a long walk back thru the museum and the longer walk around the Vatican to St. Peter's. Regards, Walter

Last November some friends of mine (one was on a wheelchair) came to visit Rome. We went to see the Sistine Chapel and then St Peter's Church. Being on a wheelchair, my friend had no problem in reaching the Church but as far as I know there are no shuttles that connect the two points. Via di Porta Angelica (the street that connects Piazza del Risorgimento to St PEter's church) is a pedestrian street. You could get the bus number 23 from Viale Vaticano to the beginning of Via della Conciliazione, but there would always be a part of street to do on foot. So the best thing would be to get a cab.
Once inside the church, go on the left and you can get an elevator that will take you to the first level, the terrace where you can have quite a good view on the piazza, but this is as far as you can go. No way to reach the top of the cupola. The first level was the point where my friend on a wheelchair and us went.
Your husband is not in such a condition (he is not on a wheelchair I mean), but the answer to your question is YES, there is this elevator that you can use easily!