third-party cookies and conversion tracking

Some vendors use URLs redirects to track clicks on text or display ads. In such cases, the typical way for such a a vendor to track if a conversion (i.e. a purchase) occurs is by setting a session cookie and then using that to connect the originating click with the conversion.

At my company, we are having a debate. Is it possible for a tracking vendor to use a URL redirect but also be able to set a first-party cookie on a user's browser? Or can a vendor who uses a redirect only set a third-party cookie as the cookie is being set on a server that is sitting in the vendor's domain and not the domain of the destination website.

1. User Clicks on a Google Text Ad
2. The destination URL on the text ad redirects the user to a vendor's server (to ostensibly track the click) and then attempts to set a cookie on the user's browser
3. The user then gets redirected to the actual d开发者_JAVA技巧estination site

In the above sequence, can such a vendor set a first-party cookie on the user's browser despite the server sitting on the vendor's (not advertiser's) domain?

As far as I know, the security settings in cookies allow a site to set cookies on their domain only. If you want to have a cookie set by a different site, you need to get the browser re-directed through that site.