Hi.
No they are not using any web service, they are creating and deleting temp sessions in the database.
As I've said we maintain a copy db but we can't do that on a live environment for reports, the users are querying at known or random times the live data so the data is changing by the second over at least 10 different databases with millions of rows.
Also I'm not sure that an approval can be made to spend money on more hardware so we can maintain something as Transactional Replication .
Discussions with the 3rd party developers are contacted but not on a database level. They are also maintaining their programs for hundred or thousands of different clients so in a way I can understand why they do not tend to support any custom database changes (even if they have messed up the indexes 
).
Anyhow, as Kristen said, if we are able to have a minimum loss of data at some custom reports, we can live with that and have the daily backup database have the previous day data that is very sensitive, at least for the accounting department.
I'm taking all suggestions in consideration but I'm afraid, currently, there is not much that can be done on the 3rd party querying let alone on changing their db structure.
Thanks again.
P.S. Don't get me wrong on "with nolock", I do not agree on using it also but I guess this is one of the situations that it can be used.