it depends - and that's usually the right answer, because it's really about whose hands the merch will end up in.
Employee swag tends to fall under HR, since it's part of onboarding, culture, and internal morale. Customer-facing merch is Marketing's turf, because it's a promotional touchpoint. But here's where I've seen the line get blurry: regardless of who buys it, Marketing should always be the gatekeeper on anything that carries the logo. i've had too many experiences where HR sourced a lovely hoodie - but the logo was a pixelated mess, placed too low, or in a colour that clashes with the brand palette. it's not about control, it's about brand consistency. Think of your logo as a signature - you wouldn't let someone forge it just because they're good at picking out stationery.
My rule of thumb: HR handles logistics and budget for internal items, but Marketing provides the official files, dimensions, colour codes, and placement approval. customer swag? marketing owns the whole process, start to finish. that way, everyone's clear on whose job it is, and the brand doesn't suffer.