Is vinyl flooring or carpet cheaper if we are the ones putting it in?

Carpet and vinyl both have a huge price range. It like asking is it cheaper to buy shoes or a coat.

Cheap? That depends, there is super cheap carpet you could lay down and not even use a pad. But it looks awful and does not last. 

There is super cheap rolled vinyl you can lay down and trim with a knife or pair of shears. It too looks awful and will not last long.

If you are just trying to dump an ugly house and want the cheapest thing to possibly do, buy a super cheap carpet, rent a carpet stretcher, and at least watch a video on how to install a carpet.

Depending on the grade of Vinyl vs carpet and it's padding, plus tacking strips.

I think you'd have better luck, installation wise and moneywise, to go with vinyl. Carpet gets stained and holds allergens, smells and pounds of dirt. Plus, it is not pet friendly either.

If you want, you can always add area rugs over vinyl flooring.

You don't have the carpet stretcher (if you are doing wall to wall carpet.)  In that case I would hire the guys who install carpets to do it right the first time.  Otherwise you room looks like schit "spread out." Lino tiles went out with the dinosaurs.  Now it is lino sheet and the floor layers do that  because that is what they do.  Lino is in the bathroom or kitchen and that is it.(sometimes stairs...it depends)

The cheapest way to go is vinyl tiles. It is so outdated. (It still may have asbestos in the tiles it is that old.)And you have to buy more than enough because the color changes between LOTS.(nothing looks worse than a patch job) So you get enough that you throw away the excess.  That applies to carpet, lino, wallpaper, roofing tiles, lawn turf, wall paint.

Use vinyl because it is easier to install.  You can make a mistake and just replace that square.  

it might be cheaper

Either one will be cheaper if you install it...

Vinyl or carpet cost depends on the style and quality ....So they both could be less expensive than the other depending on what quality you are comparing it to.

If you have never installed either one do some research on instillation.

Yes, obviously.  Carpet layers don't work for free.  Good luck following basic instructions.