This is quite old, but if time / speed is an issue, Diamond is an accelerated BLAST aligner that is very fast. I've used it with a downloaded UniRef90 database and it works both for local testing & development (2015 Macbook Pro) as well as on the production servers. I can share code with you if you want.
UniProt offer a webform for BLAST here, http://www.uniprot.org/blast/ - but this is as far as I know for use 'by hand' only rather than scripting from something like Biopython. I'm not aware of anyone else hosting a free to use BLAST service against UniProt.
The alternative would be to download the UniProt database (in FASTA format), make a local BLAST database with makeblastdb, and then search it locally with blastp (from the NCBI BLAST+ suite). Your local Linux systems administrator may already have local copies of important BLAST databases like the NCBI NR database, and could add the UniProt databases for you perhaps?
How many query sequences do you have? Even the NCBI online BLAST service is not suitable for large numbers of queries, and best done locally (on your cluster if needed).