How can I add rewards cards to wallet?

You can only do that if the rewards card issuer, like those you mentioned, support it. As for adding them it’s the same as adding a credit card. There are only a few rewards cards that work at the moment. I have my Walgreens and Panera Bread rewards cards in my wallet app.

You need to investigate whether the merchant supports the Wallet app. Perhaps someone else will come along to add to the list I just posted.

Short answer... you can't unless you try your luck with a 3rd party app.

I normally research buying new techy stuff to the 9's but I figured "hey... it's Apple... of course you can do this and that on their phones" - nope I was wrong. You can't add things that you would normally carry in your real wallet into the 'Apple Wallet'... not sure why they called it apple wallet and not just apple pay.

You could always get a Samsung phone and use Samsung Pay which will let you add ANYTHING you want... it doesn't even need to be a membership/reward card or have a bar code on it... it lets you manually enter the info you want and will even make a barcode for you and let you take a pic of it. I did this with everything that was in my wallet EXCEPT for my credit/payment cards which I carry in my phones case. I customized the app settings so I can just swipe up from the bottom of the screen and it opens... even from the lock screen if I want... simple !

It's entirely dependent upon the retailer as they need to support Apple Wallet.

That’s nice but Samsung Pay and Apple Pay are completely different (and incompatible) technologies.

You can try a 3rd party app... I think its called stocard ?