injecting chicken

You could use the brine the peppers are in and then just add the peppers into the roasting pan (with onion, tomatoes, etc.) to use later as a pan-sauce.

You might also try making a brine with the peppers and soaking the chicken in it for as many hours as possible - not more than a day though.. I usually brine my birds with a salt & sugar plus lots of herbs (sometimes just herbs de Provence though) and if it's a longish brine, the meat always picks up the flavor. I find it amazing every time!

Yes, somewhat, but not as thoroughly as injection; it is usually liquid flavoring agents that are injected. If it were me, I would make the purée (paste) and just spread it in various places under the skin especially the breasts and leg joint/thigh area

Why not puress them, loosen the skin, and rub the puree under the skin? I do this with herb pastes.

A third method (the least desired in my opinion) would be to use a paring knife and make small slits through the skin & into the flesh of the chicken & insert pieces of the peppers

Yes, the method of pureeing the peppers with one of the liquids I mentioned in the earlier post would require an injector if you want the peppers' flavoring to be injected into the meat as opposed to just under the skin. Injectors are usually quite moderately priced

You could chop them and cut skin slightly in a few spots & insert them. The only way to inject them into the meat would be to purée them in cuisinart or blender with a little liquid (their juice, white wine, broth, barbecue marinade, etc. )