We're creating a new consumer/public-facing ASP.Net web app. There are two concerns:
--Use cookie or cookieless forms authentication?
--If we decide not to use cookies at all, how would you store the data that would otherwise be stored in the cookie (Customer ID, AffiliateID, etc.). Does the ASP.Net authentication framework track something like CustomerID?
If you're doing authentication, cookies are the usual method. It's very rare these days that people will have cookies turned off because so many sites already depend on them.
Having said that, ASP.NET does support "cookieless" authentication. Basically it just adds the authentication token as a parameter on the URL. It parses all outbound URLs to ensure that they also include the token information. Personally, I wouldn't bother with this and just go with requiring cookies. There are a few additional headaches when trying to go cookieless (for example, it can make SEO that much harder, because the search engines will see a different URL every time it crawls the page).
For a normal web app there is no good reason to use cookieless authentication - Fear of cookies died out about a decade ago.
For actual data, the session object is generally a better choice than individual cookies - The session cookie is a single value that effectively gives you a key to whatever session data you have stored on the server. There are certain specialized cases where there are problems with using session, for example in multi-server deployments, but in for most applications it is simple and adequate.
The standard forms authentication system does track the username - generally this is enough to look up whatever data you need from your database if you don't want to keep anything in the session.