אני נועל סנדלים עם גרביים בחורף of course when there is no snow :-) around
Could this also mean 'They put on shoes with socks',
or might we use a different verb for the action of putting on the shoes and socks ?
Technically you need to use the verb "לגרוב" or "ללבוש" for "גרביים", so it would be "הם גורבים גרביים". But since we use the word with ("עם"), this sentence could also mean they put on shoes with socks so you are right.