Try Survey Monkey.

I'm not aware of a universal website. The best way to choose a standardized instrument would be to decide what you want to measure and then look for a systematic review of instruments that purport to measure it. For example, there are a number of instruments that claim to measure health related quality of life. There are reviews that compare the instruments and point out strengths and weaknesses of each. Similarly, a number of scales for depression exist and there are reviews that compare and contrast the various instruments. A good literature review in your field will give you some ideas as well as the language to use in your pubmed search.

Thank you Emmanuel and Evelyn.

To share your standardized questionnaire, include it as an Appendix to your manuscripts when submitting to journals for publication.