Yes!
Thatch Market is available to every member, regardless of whether your employer has EHE (eligible health expense) spending turned on.
You can browse and purchase items using your own personal card — it's a standard part of Thatch for everyone, just like your member dashboard, and it doesn't touch your employer's health budget or involve your employer at all.
The one exception is a small set of integrated partners (like Lyra and Vera Health) whose checkout uses your Thatch balance directly — those stay hidden if you don't have EHE spend enabled, since that checkout option wouldn't apply to you. For everything else, you'll just be redirected to the partner's site to complete your purchase (often with a referral or discount code).