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Maybe possible if the leaser allows it. Normally to ship you need the title or a letter from the bank allowing the vehicle to leave the mainland.

HOWEVER......I am not aware of anybody shipping a leassed car to PR.

As to taxes, you will likely need to call the tax office to check and get an extimate. Even if others have done it, your amount if any will be different.

Sorry

RE: the taxes, that's my main concern.  I haven't actually leased the car yet, I just think it would be easier to do in Miami than San Juan, plus the cars cost less in Miami.

Everything I've read about bringing a car into PR seems to apply to owned cars.

I'm hoping that by bringing a leased car with me, I can do so without paying the taxes.

I with Rey no sure if it could be done. You are not  the owner, if you stop paying  the leasing company would have to do a lot to get it back

Most banks will not let you do that unless you're a military. You will still have to pay taxes in the island since you will de required to register the car in PR.

What is the name of the government office I will have to deal with to pay taxes when importing a vehicle?  Does anybody have their contact info, and have you dealt with them before?

The calculator is at

Hacienda portal
The phone number is on the page.
Here it's the info:
Departamento de Hacienda
Edificio Intendente Ramírez
10 Paseo Covadonga
San Juan, 00901
Tel: (787) 622-0123
P.O. Box 9024140
San Juan, P.R. 00902-4140
info@hacienda.pr.gov