We have been using the following website to check active duty dates and there is a problem with the security certificate with the website and our IT helpdesk is not recommending we use it.

Is there is any other way to verify their active duty dates?

https://www.dmdc.osd.mil/scra/owa/home

https://www.servicememberscivilreliefact.com/

But it isn't free. It also isn't directly tied to the DoD database.

Most military sites (anything ending in a .mil) have security certificate issues. If you accept the security certificate one time, you shouldn't have the problem again. Obviously the most accurate way to verify whether someone is serving in any branch of the military is by running it through the Department of Defense database, which is located at the web address you have above. I wouldn't pay for this check when I can access it for free directly from the DoD.

Our IT manager has also said we are not to use this URL. I did not have success by accepting the certicate once. The second time I went in it came up with the same warning.
Andy...where does the site you provided come from?

Just research for an alternative. The site your IT wants you to refuse is the best site, to my knowledge.