I assume the DRM-free GOG release of the game lacks any daily stuff, as the only thing I see in the menu is Play and Co-op, other than the usual Settings, Stats and Quit. When you enter play, all you get is a character select screen, nothing more.
I tried taking screenshots of the game, so you could check it out for yourself, but it seems I can't, for some reason (Print Screen doesn't seem to work with Nuclear Throne). Still, for what it's worth, I'm having a blast with the game either way, even without daily challenges and missions and whatnot. Maybe they'll implement that at a later date? As far as I'm aware, the game was only officially released a month ago, perhaps they are still building things up?
Still, answering your question directly: no, as of now, there aren't any daily missions available in this GOG version.
Well to be honest, I'm not impressed at daily stuff, totally lost interest for Isaac's dailies (and I wasn't interested to begin with) but it's a bummer to have less features here...
I'll wait for a sale then.