This agency asked me to PM them through TER. How do I do that?
What do I type in the "To" section while sending mail? I tried the agency's email, the providers TER ID number, both didn't go through.

Click on the envelope under their handle.  It wil open an email that you can send.

Swim

I do not see any envelope under the handle. On the top left, it says TER ID XZxXxxx, nothing below that.

and search the ID # and see what comes up.  If the handle has no envelope under it, they are not able to get email thru the TER system.  Unless you have their public email address, sorry, don't know what else to tell you.

Swim

If the lady in question has her profile linked to her handle (TER user name), there will be a green "button" to send PM.  Click on that and it will open a message dialog.  If the lady has reviews and that option is not there, then she has not requested that her review profile and user name be linked, and consequently does not have PM privileges unless she has paid for VIP, in which case you would need to know handle to address the PM, since it is not associated with the TER ID you are looking at.

All you need to PM someone on TER is their TER handle. You write their TER handle in the To: section of a PM. Since yours is jacobsoler I would write jacobsoler in the To: field in order to send you a PM.  As you found out the TER ID# or email address won't work. Only the TER handle works.

Click on the "Mail" tab at the top of the page. That will take you to the "Mail" page. Click on "Compose" (in teal colored font) on the far left at the top of the page. The PM form will pop up. Fill in the To: field with the agency's TER handle. Fill in the subject. Write your PM and then click the "Send" button at the bottom of the PM. If you don't know the TER handle of the agency you'll have to get it from them.

Many agencies don't have TER handles but their owner or "booker" might. (In  my area many agencies don't have PM capability and say so in the TER ads. They request you call or e-mail.) In your case they are likely trying to verify you are a TER member. There are ways to find the TER handle of people you want to PM but it requires finding a post by them on a discussion board. Then click on the envelope under their handle in the post. This will bring up the PM form with their handle already in the To: field. You can search for their profile but agency profiles (in the "Site Review" section) don't have the TER handle showing.  

Since this agency asked you to PM them they must have a handle and PM privileges. I recommend you contact them and ask for the handle they want you to send the PM to.