To re-frame your question,"What is the quality of my website?"
Testing is a simple means of measuring that quality. But, there are other means that can achieve some of those same results:
- paired programming
- code review
- test driven development
So, the next question in turn becomes "How do I measure quality?"
From that becomes what your traditional testers deliver, but really anyone could deliver: business, developer, or tester.
So, what do you want to deliver that measures quality?
- customer use cases
I would highly recommend a layered approach similar to what was mentioned in earlier posts:
-user acceptance tests
In regards to user acceptance tests I would highly recommend an Behavior Driven Development (BDD) approach or a test approach similar to Specification by Design. A few example framworks around this are Cucumber or Robot Framework.