If I get a credit card without providing a SSN/ITIN, will they still report my activities like normal?

I doubt you're going to find very many banks in the U.S. that will issue a credit card to anyone without an ITIN or SSN. I did find an article which claims CitiBank might, and I've included the link to it here. Another option is to become an authorized user on someone else's credit card accounts, although that could be a challenge too without an ITIN or SSN. Since credit bureaus key their records to SSN or ITIN numbers, it's an open question how you would have a credit bureau file without either one.
Out of curiosity, is there a reason you don't have an ITIN? (maybe you just haven't applied yet?)

Here is the link to the article I mentioned, and hopefully it will give you some ideas on what else you can do:

How To Get A Credit Card Without A Social Security Number