Which spelling is correct: "Re-order" or "Reorder"

Reorder.

With re- words, you should use ‘re-’ (with a hyphen) if the next word begins with an ‘e’ or a ‘u’ (when not pronounced like ‘you’). Otherwise, don’t hyphenate. It’s therefore re-examine, re-urge, re-entry and re-elect, and reuse, reunion, reorder, reinforce and redevelop.

Source: http://www.proofreadinglondon.com/blog/to-hyphenate-or-not-to-hyphenate

In general, try to avoid putting hyphens into words formed of one word and a short prefix

Source: http://www.economist.com/style-guide/hyphens

I would imagine that both are equally viable in that context and that it comes down to which you prefer using, either visually or in terms of what you believe to be correct. If you find different dictonary/definition websites with each using or referring to a different spelling, then that would suggest that you could use either. Hope this helps.