Question though: how stable is MySQL in comparison to say, a DBM or a flat file for small (under 5000 entries) databases?
Well, that depends - mostly on the load, I'd think. With such a small db it's hard to generate enough load to really stress the system. You probably would not feel the difference at all as everything will fit in memory just nicely.
With a RDBMS you have the flexibility of the relational data model and the power of SQL. It may not be necessary in this project, but maybe in the next one. I found PostgreSQL Tutorial to be a fairly good introduction to SQL.
Is it realistic to think that I could set up a system that allows them to add, edit, and remove records with a web interface and not have to be there to maintain it?
If you set up any system - especially a database - it is bound to require maintenance. Something can always go wrong, and eventually will.
Here's one (a bit old) article which I found enlightning: MySQL and PostgreSQL Compared by Tim Perdue; and here's another: Why Not MySQL? by Ben Adida.