>Is the gargling potentially caused because by the double sink

Yes. Perhaps installing it knocked some crap loose inside the vent stack and clogged it.

>What does adding a vent look like? Looks like it is a major task

It's either relatively easy or very much not a DIY thing if you have to ask the question.

First things first, when you took out the sink in the first place was there something that looked like this? https://imgur.com/NYtQ9NS A weird vertical column topped by something with a little clearance under the counter as possible.

That's an "air admittance valve" -- you need to check with your local government because they're not to code everywhere. It's probably to code, especially if there was one there before, but it might not. Best to check. Basically, it's a one-way valve that lets air in but not out. It lets you drain properly without letting sewage gasses into the room.

If there wasn't something like that then you 100% do have a vent already. Somewhere on your roof you have one or more of these: https://i.stack.imgur.com/K3r2S.jpg That's the vent. Your drain pipes inside the wall will have a T-junction, one side going straight up to vent through the roof and the other side going down to the sewer/septic. If you have to install one yourself... that's the "very much not DIY territory." You'll have to open up the walls, penetrate your roof and properly seal it to keep water from leaking in, you have to get code inspections done. Sure, it's possible to do it DIY, but it's not really plausible unless you already do that sort of thing for a living.