The good news.
Eurostar do have a partner agreement with Avis. Although the Avis
depot is at Euston, Avis say they'll refund the taxi fee for the
transfer to St P
The bad news.
That Avis depot is not open on Sunday.
I fail to see what use this 'partner' agreement is if the depot isn't
open to meet trains. At airports, these agreements always stipulate
that the rental depot hours match the times that flights arrive/
depart. Eurostar don't seem to have thought this through...

I don't know of any car rental depot in central London that is open on
Sunday. (The Alamo depot mentioned by another poster is also closed
on Sunday).

However - the Avis depot at Ebbsfleet is open on Sundays.... That
might be the option to go for. Unless you are very familiar with
driving in central London, I would strongly suggest you should use the
M25