I am thinking it would be "je peux conduire nous au chalet" but I am unsure of the placement of the word nous in the sentence.
In declarative clauses with no inversion, object pronouns, in French, precede the verbs to which they are attached. This may be observed in phrases such as "s'il vous plaît" (if it pleases you). In that example, the verb is plaire, and its subject is clearly il and not vous; otherwise it would be conjugated as plaisez. The object then is vous, and it is evidently situated before its verb. This can also be illustrated by other common phrasebook expressions such as je vous en prie, je m'appelle…, etc.
In your example, pouvoir is a modal verb that modifies conduire, and nous is an object for conduire. In this case, nous will precede conduire but succeed pouvoir: je peux nous conduire au chalet. This applies to other constructions with semi-auxiliary verbs, like those with aller, devoir, and venir:
Elle va vous apprendre le piano. (She will teach you [to play] piano.)
Il doit vous aider. (He has to help you.)
On vient d'en parler. (We have just spoken about it.)
Note that this does not apply to constructions with the typical auxiliary verbs avoir and être, which in any case do not contain infinitives:
Je l'ai rencontrée un jour de vendange. (I met her one day while at the grape Harvest.)
Je me suis dit [qqc] (I told myself [something])
The explanations given in this answer are fine as far as basic grammar goes, but the translation is not idiomatic French. You just don't say "Je nous y conduis.", the reason, apparently is a point of logic: you do not "drive yourself" somewhere; you wouldn't say that in English in certain cases: you can say for instance "I'll drive you down to the shop." but not "I'll drive myself down to the shop." (ngram). As when you use the first person plural you include yourself, you always understate the action as applying to yourself too, and that's why, in French, the first person plural is excluded for certain verbs.
Nous pouvons aller au chalet.
context: there is a common understanding among the group of people that they are going somewhere together. In this case, the means of transportation is clear and nothing might be said about it.
If the means of transportation is not known, then precisions might be added, as below.
Nous pouvons aller au chalet, on prend ma voiture.
Nous pouvons aller au chalet, je conduis.
Nous pouvons aller au chalet ensemble, nous y irons avec/dans ma voiture.