Learn English – Am I using “suggest that” correctly

Your use of "that" is alright; however, it puts that clause into the subjunctive mood, so you need to change the verb's conjugation:

Can you suggest to them that Company Name develop the Italian version of the website?

Edit

As others have pointed out, that answer was written a bit too hastily; it was misleading in at least one respect, and I missed including some of the nuance behind what I was saying.

First, it's not the "that" that's selecting for the unconjugated verb, but rather the verb "suggests". Fair enough. Second, the sentence is grammatical as is, but for what I would (unscientifically) consider to be a significant percentage of native American English speakers, it implies a meaning different from what the OP said they intended. More on that below.

Now, the subjunctive. No, it doesn't formally exist in English as a verbal construction, and yes, referring to it that way might confuse speakers of a language that does have a special conjugation for mood. But old habits die hard, and I find it to be a useful way to choose an appropriate verbal conjugation in subordinate clauses (or in this case, no conjugation; the bare infinitive form is the choice that conveys the OP's intention most clearly).

This is why I maintain that it's a useful concept: Take a look at languages that do have a true subjunctive mood - say, French. Il faut que vous soyez ici à 11h. / It's necessary that you be here at 11 o'clock. The special conjugation of être is expressive of the implicit context of the phrase - tomorrow hasn't happened; you aren't here at 11 o'clock yet. This is how the subjunctive mood works; a subjunctive clause is used when referring to a situation that isn't the current reality. OP said they wanted their company to develop the new website, and that apparently isn't the plan at the moment, so I'd suggest choosing a verb form that signals this.

Now, linguists (myself included at times) get all cranky when confronted with "rules" mostly invented out of thin air by self-appointed "grammarians", which is likely the reason American kids in grade school back in the day (or those of us taught by people who were in grade school back in the day) were told to call this the "subjunctive" in the first place. The "experts" saw a passing similarity in modern usage to a Latin construction, and, without any sort of linguistic inquiry (or generative analysis, which hadn't been invented yet), made up a half-baked explanation and wrote it in a book.

After having learned a little more about a variety of languages and studying linguistics myself, though, I still occasionally find one of the old-fashioned explanations useful. This is one of those times.

Think of it like Pascal's Wager, but for verbs. A lot of the "rules" change over time (as they're governed by usage, not fiat), and people might understand your meaning whichever conjugation you choose, but you lose nothing by choosing the infinitive form, and for a certain percentage of interlocutors, you'll add to the clarity of your speech.