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

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.

Having under eye filler will not negatively impact your immune system. You may be on some medications for prevention of rejection. I would get a consultation with your MD before doing this. Under eye filler can cause bruising and sometimes gives over filling appearance. Make sure you go to an experienced certified reputable injector or get few opinions first, before decision making.