What size distribution of promotional t-shirts should I buy?
I think some of the largest ones. But maybe you should do some research of your market first. Buy some cheap tees from https://www.gotapparel.com/c-1-t-shirts.aspx of different sizes, make the designs. And you will soon see which of the sizes are the most needed
I used to have t-shirts made in college by a trusted source, and he was very helpful in figuring out a good ratio. Depending on your relationship with the supplier, you may broach the question to them. Depending on how big the department is you could just try and figure out what people wear (i.e. <50 staff/student probably do able).
I worked on this same problem last semester during a project helping the campus blood drive develop a new t-shirt giveaway. We ordered adult unisex shirts, and the majority of the co-ed females, and quite a few of the co-ed males got small shirts. Medium was next. Faculty/older male grad students seemed to prefer large universally and the few x-large and xx-large were left over. The style on campus was definitely for tighter shirts. We also found the shirt people to be unhelpful, but ymmv.