You can use vector graphics in the design of a website (as created in Adobe Illustrator or Flash). However they will only remain in their vector form so long as you are using them within those programs (.ai or .fla file-types).
These days, you can even place or drag .ai files into a Photoshop document (.psd) to further your website design – and the .ai file will live on its own layer as an smart vector object (it’ll maintain it’s crispness and scalability and still be editable through Illustrator).
ANYWAY…once you start moving your graphics and images into a program like Adobe Dreamweaver or want to upload them to some web-based site creation tool, they need to be outputted as .jpg, .gif or .png. All three of these file-types are pixel-based (as opposed to vector) and the conversion process will therefore rasterize (turn into pixels) any vector images contained in your design or graphics.
So…
During production and design: Yes
To output/upload to the final site: No
Vector graphics can only output in their original form in print
also, note that if you have a vector graphic that was originally designed for print (a logo for example), it’s most likely in CMYK color format. It will convert to an RGB profile when turned into a .jpg, .gif, or .png and you may see some color shift and have to adjust for that.