I've made purchases through GOG before, getting postal uncut and shadow warrior classic complete. I had no problem buying these FREE games.

I'm now trying to buy Tyrian 2000 Stargunner, and Hello Neighbor Alpha Version. I can get to the checkout, they're all in my cart, but the pay for your order button is greyed out.

It's the same for other games I try to buy. I don't have any money in my GOG wallet, nor did I when I got Postal and Shadow Warrior.

I get the same problem whether I use galaxy or in-browser.

Can someone please help?

2nd edit: Oh, known issue, but the trick to add a paid game, select a payment method, then remove the paid game does seem to work.

Are you sure Tyrian and Stargunner aren't already in your library? Because I thought those are automatically added when you make the account.

PS: If not, be aware that there are oddities with payment lately, see for example
https://www.gog.com/forum/general/your_payments_status_could_not_be_verified_due_to_temporary_technical_issues
https://www.gog.com/forum/general/i_cant_seem_to_complete_my_order_using_gog_wallet_funds
https://www.gog.com/forum/general/any_way_to_buy_a_game_on_gog_with_us_bound_visa_gift_card

Even if the game is free, may need to pick a, or a different, payment method. May also help to add a non-free game to cart, get to checkout, select some payment method that may work (Paysafecard may be the safe one), then clear the cart and add the free games you want and try again.