Our Shopify GTM App absolutely works with Shopify Non-Plus stores. In fact, many of our users are Non-Plus store owners and they see a lot of benefits from installing the app (check out our reviews).
The difference between using the GTM App on a Non-Plus store vs. a Shopify Plus store is that at this time Shopify does not allow us access to the checkout pages for Non-Plus stores. Because of this, we only recommend moving GA to GTM if you are on Shopify Plus.
However, transactions will still work as normal on Non-Plus stores and you can still use our app and dataLayer on the thank you page (purchase event). You can also use GTM to fire transaction events for AdWords, Facebook, Bing, etc.
In case you haven't seen it, here is our guide which has step by step instructions (including a video!) on installing the GTM App on Non-Plus stores.