There are a few Latin verbs ending in -izare, and they are almost all Greek loanwords. (This list was generated by using the search for “words ending with …” on the Perseus server—do note that it contains a few false positives, some of the entries are no verbs at all.) And this is not surprising, as -ῐ́ζω is a common suffix in Greek used for forming verbs from other words. When Latin speakers imported these verbs, the Latin forms ended in -izo (first person singular present active), and it must have appeared natural to them to put these words in the first conjugation. The infinitive -izare is a purely Latin form.
Usually the suffix was imported into Latin as part of the whole verb. But was this, or did this at some point become a productive suffix in Latin as well? A hint would be if there were purely Latin words formed in this way. (Another hint would be Greek imports where the noun or adjective is attested in Greek but not the -izo verb.) Looking through the previously linked list, I see two: pulverizare and latinizare. Both are Late Latin and rare forms. So strictly speaking, yes, words were formed that way in Latin, but realistically, no, it was not really a productive suffix.
But this particular question need not really concern you, as synchronus is, of course, a Greek loanword anyway (from σῠ́γχρονος), and the verb σῠγχρονῐ́ζω does in fact exist. (Its meanings are probably not exactly what you have in mind, but let that not distract us.) So synchronizo (infinitive synchronizare) would be a perfectly good way to import the word into Latin, even if it is not attested in Classical or Late Latin.
Having a first-conjugation verb synchronizare, it is perfectly straightforward in Latin to form a noun synchronizatio, -onis, f.
But was this really done? Again using the suffix search, we find very little. Mostly it is two words: baptizatio “baptism” from baptizare, and gargarizatio “gargling” from gargarizare, and then there is the funny bombizatio “the buzzing of bees” (which certainly derives from Greek βόμβος). So this does not look like it was a common thing to do, but then one has to keep in mind that the list of verbs ending in -izare is already short and contains a number of rarely used words, so what do we expect?
Having said this, the Lexicon Latinum Hodiernum by Peter Lichtenberger offers synchronizatio as a translation for German Synchronisation (referencing another modern German-Latin dictionary, the Lexicon Auxiliare, from 1991).
In my opinion it makes little sense to use synchronus facere (or synchronum reddere, which the Lexicon Latinum Hodiernum also offers). By using synchronus, you have already lost the game of finding a particularly “Latin” expression.
To find a more Latin way to express the idea, you have to think about what you are talking about specifically when you say “synchronization.” For example, the L.L.H. offers another translation: transmutatio vocum (ad taeniam cinematographicam). How in the world did they arrive at that? Ah, you see, in German, Synchronisation (among many more mundane other things) is the word for what our English-speaking friends call the “dubbing” of films, that is, replacing the spoken words in the sound track with a translated version. In a similar manner, you may find another way to express what you are referring to.