Actually, “any” can have multiple translations depending on the meaning of your sentence.

Your sentence has a negation in it, so “any” would translate to “aucun”, which means “none”.
But if your sentence was positive, it would translate to “n’importe quel”.

Full sentence is: “Je ne puis le trouver dans aucun dictionnaire.”

It’s common to say:

Je ne l’ai pas trouvé dans le dictionnaire.

It’s singular and uses the definite article so it literally means “I did not find it in the dictionary” but I believe the French meaning is a little more ambiguous and you could still use it if you tried several dictionaries.

Speculating a bit, French speakers tend to consider that there is only one correct way to use the language. There are in fact several dictionary publishers making slightly different choices but we still say “le dictionnaire” all the time without specifying which one. The implication is that if it’s a “real” word, then it should be in all of them, we don’t have a clear concept of competing norms for the language (cf. also all the mythology around the role of the Académie française).

If you really want to explicitly stress that you looked at several dictionaries, then you have to use “aucun” as Nathan explained. However, “je ne puis” is extremely formal/outdated. You simply cannot say that without irony today.

A better translation for “I can’t find it in any dictionary” is therefore:

Je ne peux le trouver dans aucun dictionnaire.

It still sounds quite formal, not necessarily as neutral as the English sentence. Using the past tense sounds better to my ears:

Je n’ai pu le trouver dans aucun dictionnaire.

Also, in French, it’s usual to say “I did not find it” rather than “I can’t find it”. A less faithful but even more idiomatic translation would therefore be:

Je ne l’ai trouvé dans aucun dictionnaire.

Finally, “n’importe quel dictionnaire” is actually correct but does not work in the negative. You can however say something like

Tu le trouveras dans n’importe quel dictionnaire.

Literally: “You will find it in any dictionary.”