I found out how to do what I needed done; it's not quite "runtime" but close enough and I didn't have to modify my machine.config. I added this section to my web.config file:
<system.data>
<DbProviderFactories>
<add name="Npgsql Data Provider" invariant="Npgsql" description=".NET Framework Data Provider for PostgreSQL" type="Npgsql.NpgsqlFactory, Npgsql" />
</DbProviderFactories>
</system.data>
...and it started working like a champ.