Refinancing itself shouldn't hurt your credit score. When you're simply shopping for a loan, lenders may pull a “soft” credit inquiry, which won't affect your score. Once you actually apply, Webster Bank will request your full credit report. That's called a “hard” inquiry, and it may drop your score by just a few points.
Multiple inquiries to different student loan providers over a short period of time are considered one inquiry. Therefore, they won’t have much impact on your score. However, if you make multiple applications for refinancing, each lender will pull a “hard” inquiry. That can have a negative effect on your score.
And of course, if you fail to make payments on your present loan while waiting for approval on refinancing, your credit score will fall.