The databases and software can be downloaded and run locally. I regularly do several hundred proteins through my local install with some additional scripts to agregate the results.
I am biased since I am the developers of MMseqs2. MMseqs2 can perform fast sequence/profile searches. It is possible to annotate 1.1 billion protein sequences with Pfam domains in 8.3 h on a 2×14-core server. Here is a guide how to setup the search.