I ordered a laser hair regrowth cap that I want to start using. Will it have any side effects on my baby?

It  is not known whether low level laser light therapy is safe during pregnancy. Some Studies have shown that treating the scalp is a sufficient distance away from the baby. However, since low level laser does produce chemical changes in the scalp, I would not recommend it during pregnancy. Most likely, the changes of pregnancy will improve your hair and Laser cap will not be necessary.
PRP hair md.com

I am not aware of any studies on this but I would advise my patients against it.  Pregnancy in itself has tremendous hormone changes many of which can affect hair loss and changes in hair quality.  Many of these changes will probably improve after your pregnancy.  Enjoy your pregnancy and wait until after you give birth, then see what happens.

Although it is unlikely to be a problem, since there are no studies done on use of light laser caps and pregnancy, the actual effects would be unknown.  It would be advised to check with your ob/gyn on this matter.  good luck!!