Bethesda announced at E3 that PC mods for Fallout 4 will be playable on the Xbox One as well. In a follow-up interview, director Todd Howard says that the PS4 version could support mods as well.
In his interview with DigitalSpy, Howard said that they're hoping to do it. However, the decision is ultimately up to Sony:
We will be. Well, we're going to make every effort to. If Sony says no, we won't, but I don't think they will.
It's hard to imagine Sony turning down mod support for the PS4 version. Fallout 4 is going to have a huge launch and Sony wants to maximize the amount of copies sold on their console. They'd be shooting themselves in the foot if they let the Xbox One be the only console that can play mods. That feature would be a huge advantage.
I guess the only reason Sony might turn down mods is if they're worried about the game's stability. Third-party modifications can have unintended effects on a game, especially if the user is running more than one at a time. I don't think Bethesda can guarantee that every fan creation is going to work perfectly for the game so that might give Sony pause.
Still, now that PS4 owners know that Sony's the one who can decide for or against mods, it's tough for them to say no. There's going to be a lot of pressure on them to agree to mod support. Yes, it might result in some crashes but I think players will be comfortable with that risk if the trade-off is hours of extra gameplay from mods.
No matter what platform you're playing mods on, Bethesda says they'll be free. Howard confirmed in a different interview that they're not planning premium mods for Fallout 4. Bethesda caused no shortage of controversy earlier this year when they began allowing mod authors to charge for their creations on Steam. The backlash and shutdown of the feature will probably ensure that Bethesda (and Valve) doesn't try it again anytime soon.
Fallout 4 will debut this November on PS4, Xbox One and PC.