I know I will not be supplying enough information for everyone.

An auditorium/sanctuary is fairly square, with about a 35 foot high vaulted ceiling, tongue and groove. There are reverberations everywhere.

I would like to cover a portion of the back wall, someone recommended 3 to 4 inch foam, but I wonder if there is something from Amazon or somewhere else that you have used successfully that is absorbent.

Are you looking for something cheap? Search for acoustimac, however, an auditorium would need a lot of panels. Plan to cover a minimum of 50% of the surface area.

Alternately, a cheap solution could use 2" rigid fiberglass with wood firring as a frame, and cover with a fabric stretched over the firring frame (poor mans fabri-track)

Or, you could use a low density fiberglass batt, or duct liner and cover with thicker cloth like a curtain.

A proper choice would use an architectural product like:

Fabri-track
Poly-sorb
F-Sorb

Is this for speech, live performance (at ANY SPL), mixed use....?

Hire someone. Seriously. Find someone who does this as a profession- treating the back wall will help, but it can cause the problems form bare side walls to rear their ugly heads.

A whole industry for acoustical treatments exists- it's not just foam or fiberglass, either. The sources can be found by doing a search for 'acoustical product suppliers', like this one-

[Link: continentalflooring.com]

A facility that is used for speech requires specific reverberation and delay times that prevent reflected sound obscuring/masking what was said before it reaches the listener's ears. The decay rate should be natural, you should have no distinct echo and it shouldn't be totally dead. Since sound travels in three dimensions, you'll need to consider the walls, floor and ceiling.

Speech primarily, some vocalists.

I would be asking if the vocalists will be accompanied, by whom and with what instruments. If you think scope creep in a project is bad, wait until you do a project where they don't tell you everything and then do something outside of what was discussed, like describing the use as "speaking with an occasional singer, accompanied by acoustic guitar". The latter usually ends up being an amplified singer and acoustic guitar because the people in back can't hear them when un-amplified and then, the higher SPL causes problems that would never happen if they had stuck to the design parameters or if they had just volunteered the information that was so badly needed in order for the project to come close to being a success.

I installed sound panels from Perdue Acoustics at the performing arts center I run. They come with mounting brackets, they look great and dramatically improved the sound. They offer online training to better understand how, why, and where. Strongest recommendation.

Fiberglass with a beveled edge (the photos on their site show those, without looking further)? Looks like the panels I saw when a rep from a company that sold panels came in when the store where I was working wanted to have a home theater room but it was too small and suffered from terrible flutter. I looked at the panels and asked about how they were made- Owens Corning 703 with the edges soaked with resin before beveling and covering with Guilford fabric.

The old rule of thumb I used to see was that roughly 30% of the surface should be covered if measurements won't be used. Is that close to what you used?

Actually, I ended up with 2' x 4' panels off the wall, down from the ceiling. I had them on the side walls and the back walls, and they worked very well. Folks never think about budgets for this sort of thing, and really they should.

My living room had some problems that were really annoying, so I brought the panels in that I had made for a home theater Phil (old member of RC) and I did- rather than place them and hope they worked, I used Room EQ Wizard in real time, to watch the frequency response change as I moved the panels. That's a free program that not only has RTA, it has waterfall plotting, so the time component can be seen as well as the energy. Using a repeating percussive sound, starting with zero treatment, would be a good place to begin- compare the reflections/reverberation after placing some of the panels and it will be easy to see how well it's working.

[Link: ccrma.stanford.edu]

[Link: akutek.info]

If you want to really hear how bad the room is, plug one ear- you'll hear things that can't be easily heard when both ears are used. The second ear and our brain merge the sound and perform functions that are very difficult to replicate with electronic equipment.