I believe I've come up with a method to avoid the quota. For even further clarification, the quota was being hit because of the amount of users using the site and reaching the limit for how much Google allows. I think I have discovered a method for overcoming this and thank you for your patience.
Feel free to email me about any future issues.
EDIT: Further clarification: I DID NOT ADD THE QUOTA. GOOGLE IS DOING IT BECAUSE THERE ARE A LOT OF REQUESTS. I AM TRYING TO GET IT FIXED. THANK YOU.
I received all of your emails... and there were a lot.
The reason the quota limit was being displayed was because due to huge increases in the website's popularity, it was reaching over 45,000 searches per day and listing 20 videos, leading to >1,000,000 API requests to YouTube per day, which is the limit for the API.
I've temporarily decreased the videos listed to 10, but if the API reaches the quota again, I'll come up with a different workaround.
Thanks for your patience,