My initial reaction was exactly the same as Rachel's and Ghol's. AFAIK I use 'resort' in this noun sense only in the phrase 'last resort.' The sense is indeed listed in dictionaries, but in my experience it's formal and not at all common. So I'd be inclined to mark 'resort' in this sense [formal] in LEO.
Instead of 'without/by resort to' I would use 'without/by resorting to.' When the gerund is not possible, as in 'have resort to,' I think I would simply use the verb 'resort to,' or perhaps 'have recourse to' or even 'have to resort to.' I think most AE speakers would do the same (and Ghol suggests it's not just AE), so I would suggest that LEO consider adding those alternatives if they're not already in.
*without resort to - 34,700
without resorting to - 159,000
*by resort to - 11,800
by resorting to - 40,200
*have/has/had resort to - 2,940
have/has/had recourse to - 111,000
have/has/had to resort to - 198,000
It's hard to contrast 'have resort to' with 'resort to' alone, since the latter is a subset of the former, but I think most native speakers would agree that the verb is far more common than 'have' plus the noun.
In each case, the noun sense of 'resort' (marked with an asterisk) is notably less common and (if you skim the hits) more formal-sounding.
It would be even better if we could figure out how to phrase the entry in a more general way, not just as an example (i.e., lose the noun). But my brain is too tired tonight to tackle that.
@Ghol: This is a picky thing to ask. But could you possibly mention at least Collins (the bilingual one is aka HarperCollins, Pons-Collins, Collins Large German, etc.) instead of just wordreference.com? I have this thing about liking to know the actual source, and I have the feeling it might also be helpful for others to be able to tell at a glance that it's an online version of a very reputable printed dictionary, backed up by the full range of detailed editorial oversight and lexicographical research.