Here are the dictionaries I usually check:
American Heritage Dictionary: No entry.
Collins Dictionary: No entry.
Random House Dictionary (dictionary.com): No entry.
Macmillan Dictionary: No entry.
Cambridge Dictionary: No entry.
Longman Dictionary of Contemporary English: No entry.
Oxford Learner's Dictionary: No entry.
archaic : publish
Oxford Living Dictionaries:
rare with object To publish; to make publicly known, expose.
Wiktionary: 1.(rare, transitive) To publish or make publicly known.
(Note Wiktionary often features entries that other more "official" dictionaries don't. Also note the similarity in definitions between OLO and Wiktionary, one taken from the other or just coincidence?).
It has an entry on Urban Dictionary (considered a slang dictionary), with its top definition having an up-to-down vote ratio of 16 to 6, which I consider bad.
There are other dictionaries which are subscription-based (ie., you pay money), the best probably being the OED, which unlike most other dictionaries is a historical dictionary which will give more details about a word than probably any other dictionary. Unfortunately I don't have a subscription both because I am cheap and don't have a library that gives its members a free subscription.
The fact that you have seen many results on GitHub may simply be due to the fact that it's used by a certain sub-culture of people or a jargon word that's used in a particular field. For example "performant" isn't listed in most dictionaries, but many "computer people" are known to use it. Another possibility could be this is due to to a phenomenon you see from non- native English speakers. In this case a backformation of an "-ation"-ending noun to create the verb is attempted. I'm not saying this is the case with publicate, as publicate may have a history in older native English. Other words like this that appear in some dictionaries and not in others are words like configure/configurate, compile/compilate. If you do a search for any of these words ending in "-ate" where there's an established noun of the "-ation" form, you find plenty of results in English. That doesn't mean they're necessarily common or used or even recognised as words.
Either way I guess if you use "publicate" you will receive strange looks, unless of course it has a common and understood meaning in the GitHub world.