"Yes" or "No" field on provider profiles. For "Yes" entries, provide a field for us to select a percentage of donation - I'd suggest increments of 10%.  

Anyone could search and exclude (or include?) providers requiring significant deposits from searches.

I think many would search / screen on "Required = No" to rule out "Required = Yes." Requested has some vagueness to it. "She requested Dep ("yes") but one of my refs was a friend of hers..." "She requested Dep ("yes") but we agreed to a prelim skype screening ..."  
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Also, some have different touring vs local requirements: touring appointments require (requests) Dep, local appointments don't.
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I hope others will reply to help to fine tune your suggestion to avoid false negatives.

This would be immensely helpful on the provider end as well as it would cut down on inquiries from potential clients who then balk when a deposit is requested (even though they agreed to the deposit policy when filling out the booking form).  

 
As "impposter" suggested, there should be some nuance allowed. Possibly have the overarching question/statement be "Deposit Required" or "Deposit Info" and options could be "YMMV", "When Touring", "New Clients", "Dates 2H+", etc. and then next to each possible option could be the deposit percentage. I think the section would look best in the left sidebar along with the "Rate Information". Which I would also suggest streamlining the "rates" section of our TER profiles as well. In my head, the deposit section could simply be like this for example:

 
Deposit Information
This provider requests/requires a deposit under the following conditions:

New Client: Yes - 25% deposit requested
Returning Client: YMMV
Touring/Visiting: Yes - 25% deposit requested
Exclusive Travel Date: Yes (50%) + Travel expenses