Can I make citations appear as footnotes?

The @citation syntax is actually a ref.

#show ref: footnote

will make all references as footnotes, including references to labels.

The documentation gives a more detailed example
here: Reference Function – Typst Documentation
which you can adapt to result in

#show ref: it => {
  let cite = ref
  let el = it.element
  if el == none {
    footnote(it)
  } else {
    it
  }
}
#set heading(numbering: "1")
= Heading <label>

Citation: @citation

Label: @label

#bibliography("bibliography.bib")

In most IDEs, you can inspect the value of it, which is the input to the show rule on ref. Otherwise, just print it using repr.
For example, I have
image