I'm pretty sure that the problem must be with either my pick-ups or my input, but when I play a couple notes on my bass, it sort of mutes out and it sounds really quiet and distorted. I'm renting it and it has insurance but should I be worried about my guitar or my amp?
It could be your chord. Try a different chord, do you have extras or maybe a friend with a working one?
Yeah, I've tried 2 cords now, and they work fine with my electric guitar. But its similar to that kind of problem when a cord is bad, but you can still hear the notes but they're really quiet and distorted like a fuzz bass.
Check everything before coming to conclusions:

1) does youre bass use a battery? mine does and i freaked out today when i thought i blew my amp but i really just ran out of battery life.

2) is your input jack well soldered? this has happened 2-3 times on my squier

3) Faulty soldering in cables, or just badly treated cables are also a really common problem, and you'll get unwanted crackling alongside. When buying cables, dont go for the expensive ones, look at like 5 different ones WITHOUT Looking at the pricetag, open each up and see which one you feel is the most solidly built, and get that one, regardless of price. my best cables are 10 footers that cost me 4 dollars each, and ive gone through 10+ DIFFERENT BRANDS of cables at the store. dont fall for gimmicks like "gold plated" or w.e, as 9 times out of 10, that just drags up the price with very minimal or NO help in durability / anti noising, etc.

4) Use a different bass on your amp (i forgot to say this, this should go first) as this will tell you if its your bass or amp.

5) if its your amp, smelll it to see if it smells burnt.

6) Make sure you try different plug sockets for your amp, it may be a faulty socket.

7) Check if its overheating.