So, I can guess from the words 盗む "steal" and 飲む "drink" that she drank secretly without anyone knowing, however I can't find this word in a dictionnary. Why is that ?
Compound Verb Lexicon recognizes only five verbs that start with 盗み, namely 盗み聞く, 盗み去る, 盗み出す, 盗み取る and 盗み見る. However, as a noun formed from masu-stem, there are several more words that have 盗み. For example, 盗み笑い, 盗み撮り, 盗み食い are relatively common "nouns" that also work as suru-verbs. 盗み飲み is not particularly common, but anyone who knows 盗み食い can instantly guess its meaning.
This is a compound verb. We don't quite do things this way in English, but it's a bit like "
[VERB]" -- the "main" verb is the second one, and the first verb tells us more about how the second verb is being performed. Pretty much any verb can be compounded with any other verb, but most combinations won't make much sense. :)
From what I know, it was popular back then to post a cat or dog drinking something from a cup or a glass while filming it secretly.
I saw many Japanese people posted, for example:
Most probably it's an invented word, that's why it's not included in the dictionary. I can't even find one in slang dictionary.
However we can find in goo.ne.jp:
You can replace 食べる with 飲み
Stealing by drinking secretly/sneakily hidden from human/other people (secretly take a sip/drink/sneaking a bite (for food) or snitching it).
Disclaimer: I'm not a native Japanese, so it might have other nuances that I know nothing about.