I am a lung transplant recipient and want to get some fillers under my eyes, would this have any impact on my immune system?

This shouldn't negatively impact your immune system. I recommend getting a formal evaluation with a cosmetic dermatologist. Filler results take 1-2 weeks. Best, Dr. Emer

Hyaluronate-based fillers shall not adversely interact with your immune system; so, you may proceed without undue concern!  
My only advice is that you should go to a physician who observes antisepsis and works in a clean environment, in order to minimize the risk of infection. 
Also, make sure you tell the physician (it certainly doesn't matter if it's a cosmetic physician/ surgeon, family physician, dermatologist, et c.) that you're a recipient of organ transplantation and the details of relevant recent blood tests.