I have seen the word wont (short for will not) spelt as wont and won't. Which is the correct way?

All the contractions of 'something-not' should have the apostrophe - won't, can't, shan't, didn't, haven't, hadn't etc. The apostrophe is in place of the dropped letter.

Wont (without apostrophe) is a different word altogether, meaning habit or normal behaviour. 'He drove fast, as was his wont.'

Many people, including the great Irish dramatist, George Bernard Shaw, have wanted to reform English. Entirely new alphabets have been suggested, on the basis that - as things are - 'ghoti' may be pronounced 'fish'...that's 'gh' as in 'rough', 'o' as in 'women' and 'ti' as in 'nation'.
The apostrophe was one of Shaw's particular bugbears and some editions of his plays were printed without them. You might very well find the word 'wont' in such circumstances meaning 'won't' rather than 'habit'.

"Won't" is the correct way. It must have an apostrophe to mean "will not." Otherwise, as Ewood27 said, it is an entirely different word.