It's important to make sure when you get involved with another company to make sure what their terms are and have a definite copy of the project agreement to back up your end.
While a lot of these places may have their own account to conduct their work instead of a login plan for individual clients, you won't be able to get information from them unless it is in the project agreement. You can't make them do anything. This usually means that their method of working is flawed and there seems to be no drive to give stats to the client.
If they can't provide stats you may have to cut your losses if they refuse and decide to sabotage your business. If they do sabotage your site's listings that you paid for, and if you paid using a credit card, reverse the transaction and put a good case to your credit card company with proof the agreement and any other relevant information.
You can't be blackmailed by them, and you have a lot more power on your hands than you think. You can go to business service review sites and leave reviews. You can report them on scam sites too.
AND... If it really is worth your time, you can take them to small claims court if there are enough damages to discredit your business that you have to spend money to fix it.
This is definitely a tough call to make.
As for determining how many people you have converted, here are just few things to decide if you are: how many people have scheduled an appointment through the website? Are you asking new patients who called on who referred them to your business? Do you have an active newsletter and have you had new subscribers? Do you have any social network handles connected with your brand that you are engaging with new people?