I would lean towards museum board but in an Art Design course series, where the work was required to be mounted on a rigid back board not just a sheet of paper, I tended to actually use foam core. Museum board would be pretty good but also depends on the board.

Bristol board are more like heavy card stock like you might use for a post card or paperback cover on a book. This would be inadequate for my class.

Museum board would be better than Bristol as it is usually a slightly thicker board. 

When I used foam core and heavy stock photo paper (which had about the thickness of Bristol) and adhered it to the foam core which already has surfaces I can mount to that were roughly equivalent to bristol or museum board so combined was effectively the same.

My biggest issue of concern when using any light weight paper like ordinary printing paper or mild art paper is glue adhesive bleeding through if you use any thick amount of glue. Which would dock grade for poor craftsmanship. Other cases, is bubbles. I would use spray adhesive and a careful applying. I prefer the photo paper because glue would not bleed through and rarely would I have a case of bubbling. I can imagine the hell in doing that with vellum.

If I was having a more expensive option, gatorboard would be an option but given it uses plastic instead of the paper surfaces, it would make for cutting the boards trickier and harder on the razor blade of the box cutter. Alternatively, I could choose MDO or hardboard (masonite) but on a student budget, it wasn't practical to spend that on assignments every week or two. In its case, cutting it would belong in the domain of a Jeweler's piercing saw and very fine sanding and only the razor blade to trim the paper sheet to size just right.

This is my experience in one of my classes. I wouldn't assume all this is necessary for your class. 

It is probably something excessive unless you want to do a final presentation with higher than normal standard. 

What is universal and conveyable is there is an importance to craftsmanship and presentation skill quality but unless the instructor is picky for it, I wouldn't necessarily waste time and money on the inconsequential work where the cost and time would warrant little gain.

In my course, the quality of work in the regard and presentation would make entire letter grade difference like A to B because you were cheap or you were lousy and sloppy in how you mounted the paper, cut the board or paper, etc.  If the work and time has merit and you have the money, go for it.

yeah... big point is there are other additional options with more expense at that which I wouldn't recommend doing if it doesn't make any real difference in grade for assignment. In other words, if the work needed to be done doesn't warrant it then don't waste the time and the money by going overboard. Sure, might look impressive or great but if other students would get A's without going to that expense and expending that time which could be used in doing the actual work that is being graded on... such as the design....be wise and work smart and devote the time and energy in the right areas at the right time.