I get skin tags on my armpits from shaving. They've never bothered me and I've never bothered trying to remove them since they're not seen. Yesterday, I noticed I was developing a new one and when I woke up today, it was all red and itchy. Does anyone know if it's possible for them to get infected or is this just maybe a part of one growing? I feel insanely disgusting if I try scratching it.

I'm pretty sure they can get infected... I don't really know. The only experience I ever had with them where we ask professional advice was for a dog I had once, and when dogs have then, they can get infected, so I assume it's the same for humans though.
But if it's not PAINFUL or warmer to the touch then the skin around it, it's probably nothing to worry about. Icthing is usually more a sign of HEALING than infection itself, so it could have been slightly infected and getting better, or it may just be icthy (during the winter I icth EVERYWHERE, I suppose from dry skin, though some of it doesn't necessarily even look or feel dry). Maybe try some lotion just in case it's dry skin, or if it really bugs you, something like lanacane, which is personally my favorite icth cream because it's got something in it that sort of numbs the area. Like not severely so you can't feel it, but just enough so it won't itch.

They're not really harmful.

I've heard tales where you can wrap a piece of string around it at the base to get it off.

You could also try putting some ice on it to numb it and pull it off yourself.

Or you could just let it fall off.

I had one by my armpit not so long ago. I ignored it (I don't have a high pain tolerance). It would hurt if I rubbed something against it (like my bra), but not too bad. Within about a week or so it just fell off and the skin was a little red and it healed within a day or two

If I remember right, it's just loose skin or something that got pushed up and so now it has to fall of. Are they right where you shave? You could try changing the razor blade (dull blade causes razor burn), a new shaving cream (I personally use the Venus things with the soap on them, don't use shaving cream, just need water), or shaving less (I shave every other day). You could be irritating the skin. A new deodorant might help as well (I use Dove Ultimate, makes the skin smoother after a few months while shaving. Been using it for two years and I love it).

Just don't shave over them! That hurts.

My guess would be since you get them when you shave, they're mostly ingrown hairs. I've had a few male patents who have gotten them on their faces, and we've just given them bacitracin (since these ones were open, to prevent infection.) Worse case we've plucked the hair out with tweezers and it went down like normal.

The hair inside the tag was probably growing and got curled up underneath causing your skin to react, you should be okay. If it continues longer than a few days or gets bigger you should see a doctor.