You will need to setup a 302 temporary redirect from support.oldsite.com to newsite.helpshift.com. Test this first, and if everything is working properly, change it to a 301 permanent redirect.
Once you make your 301 permanent redirect, your site’s reputation, credibility, and other SEO value data saved by search engines, will be transferred.
Note: By testing with a 302 first, you can revert back without consequences. Otherwise, you may risk confusing search engines if too many modifications are made to a 301.