WASM allows arbitrary code execution in an environment that doesn't include the DOM... however it can communicate with the page where the DOM is available, and it's trivial to setup an abstraction layer that gives you the full suite of DOM manipulation tools in your WASM space. Libraries for WASM development generally provide that for you.

For example here's SwiftWASM:

let document = JSObject.global.document

var divElement = document.createElement("div")
divElement.innerText = "Hello, world"
_ = document.body.appendChild(divElement)

It's pretty much exactly the same as JavaScript, except you nee to use JSObject to access the document class (Swift can do globals, but they are generally avoided) and swift also presents a compiler warning if you execute a function (like appendChild) without doing anything with the result. Assigning it to a dummy "underscore" variable is the operator in Swift to tell the compiler you don't want the output.