Or is this shop just for wizards and witches?
Anybody can shop there. However, it's in Diagon Alley, making it hard for Muggles to get there.
Muggles can be in Diagon Alley, as evidenced in book 2 when Hermione's parents show up there:
[H]e was distracted almost at once by the sight of Hermione’s parents, who were standing nervously at the counter that ran all along the great marble hall, waiting for Hermione to introduce them.
Other than the slight barrier of a) not knowing of its existence and b) not having access to it, there's no reason why Muggles can't shop there. In fact they have a line of Muggle magic tricks:
Harry left Hermione dabbing her black eye with paste and followed Fred toward the back of the shop, where he saw a stand of card and rope tricks.
“Muggle magic tricks!” said Fred happily, pointing them out. “For freaks like Dad, you know, who love Muggle stuff. It’s not a big earner, but we do fairly steady business, they’re great novelties.
Yes, but they'd have to find it first
The average Muggle would not know about Diagon Alley, and since they can't see the entrance, couldn't stumble on it accidentally. However, there are certainly Muggles who do know about Diagon Alley, and could conceivably want to shop there.
It's not known exactly how many Muggles know about Diagon Alley, but they all would have close connections with a wizard, like the Granger family or parents of other Muggle-borns. In addition, they would be unable to see or activate the entrance by themselves. Since the entrance to Diagon Alley can only be opened by magic, a Muggle would need a wizard to open it for them.
Since the barriers and protections on Diagon Alley are designed to ensure that only wizards and "approved" Muggles can get in, there's no reason to test if someone is a wizard or not after they're already there. Once they are in Diagon Alley, it's proof enough that they have magic, or are at least allowed to know about its existence.
But if a Muggle was in Diagon Alley and could pay in the proper currency, there's nothing stopping them from shopping in the Weasley twins' store - or pretty much any other wizarding shop except for probably Ollivander's, since wizards usually try out wands before buying them and a Muggle would be unable to get a wand to perform magic.
Even shops selling items that are only useful to wizards wouldn't have a "magic check" in place to stop Muggles from buying them (unless the wizard would need to try out the item, which would incidentally act as a type of "magic check"). The items might not be any use to a Muggle, but a Muggle already in Diagon Alley could probably buy them if they want to. Why the Muggle would want something they can't use, that's a different story - maybe as a gift for a wizard they know, or maybe because they thought owning whatever it is would be cool.
Another reason why a Muggle might not be able to buy something from a shop is if the owner decided to cast anti-Muggle spells and protections on their shop specifically. The Weasley twins are unlikely to have done this, as they would likely see no reason to.