If the intention is to shift each element one position towards the front and put the first element at the back then the code looks fine. That is assuming SCORES_SIZE is equal to the size of the vector. You might not actually need SCORES_SIZE in your program at all because you can easily get the size of the vector using the size function: oldScores.size()
Note that C++ has a function template called std::rotate that does exactly this.
i dont think my teacher would accept this becuase we didnt learn the erase function
Get over this mentality. Your professors can't teach you everything; I learned c++ decades ago and back then the rule of thumb was 5 HARD years in the language to master it. You get 1, maybe 1.5 years in college, and that only an hour a day.
Learn everything you can, even if you can't use it in the class, absorb it.
And your professor is more likely to be impressed with your initiative and willingness to research and learn than whether you can do it the hard way. The hard way is good practice, but its not good in practice :)
I agree with jonnin. Back in the day when I was learning "C" I was always doing something that was not thought in class and not only was my teacher impressed it was not a problem. Learn everything you can now because in the future you may not have the time to spend on learning.