You do not say it in your question, do you have a car?
First, for the close experience, I went during August to Greece and did not book ahead (we were pedestrians). On a very large car ferry between Athens and some islands, booking the day before let us without reserved seats, but cabins with beds were still available. We went for non-reserved seats and it was less comfortable than the reserved ones for sleeping, but these are comfortable for sitting even on long distances (Santorini to Mykonos seems to take 3 hours). The seats are first-come, first-serve but even in August they probably do not sell out. So if you are pedestrians, that could be fine.
So it was not full but booking ahead would have been smarter, especially because buying tickets there made us lose time (the harbour is 1 hour from Athens center) and made our plans risky (if there were no more seats, our itinerary would have been messed up).
We took other more local boats and we never had any issues, but these were intended to be booked last-minute anyway.
As you said in comments, websites selling tickets do not seem particularly trustworthy, that's a good argument. But when I went to the harbour in Athens, there was no distinguishable "official" ferry ticket seller for a carrier, only a string of many travel agency offices selling tickets for any ferry operator. And the one I bought my ticket from made a typo when copying my name from my ID (I don't have a "greek-friendly" name, though).
Whom to book from?
On the website of Hellenic Seaways they do not mention any office where they would sell their tickets, but a forum thread suggests the travel agent Pelican Travel is the agent that is "official" reseller of Hellenic Seaways on Santorini island.
I found an article on tripadvisor on the topic that mentions a couple websites to book from. I could not find reviews for these though.
Prioritization of constraints
But overall, it is this kind of question that depends mostly on your prioritization of constraints: do you have a car? Do you have a tight schedule?
- The price does not change with time, so in our society paced by revenue management, you do not have the pressure to book early to save money
- I would say the most important criterion is how flexible you are. If you are on a trip where you gather your options in advance but make the decision of where to go next on day-to-day basis, it is preferable to wait. If on the extreme other hand, you have planned everything already and you can name every thing you do on every day of your trip, then it is better to book ahead.
- On the close topic of planning, when and how would you book your tickets if not online? You may book it at the harbour the first time you arrive, but maybe booking online beforehand is competitive. In any case, if you decide to buy tickets at the harbour, I would recommend to write down now on a paper (or your smartphone) all the options you searched, the schedules, prices, and which one you prefer and for which reason. You would not want to lose time searching again for everything the night before you go on the boat.
- Why would you not book in advance? Do you have super high fees on your credit card that makes buying it before going? I see they ask for 10 euro of reservation fee if you get your tickets there, which is low compared to the 60 euro *8 (people max per reservation) of the tickets. The site does not look trustworthy but in my experience, neither do the travel agencies in Greece.
- A search of ferries for this trip on a random day in June showed 6 possible rides, so the risk, if you have no car, of missing a boat is not too bad, you can probably board another boat during the day.
- If you have a car, you have more risks to have no spot on a boat (especially with 2 or 3 cars, as the Highspeed 5 of Hellenic Seeways has room for 90 cars). If you are a pedestrian, it is easier to "squeeze" (not literally, you will have room) 10 people on a boat.
- tickets for ferries have your name on it but it is not as strict as airplanes (for pedestrians at least), my name was wrongly spelled and they did not care.
- Santorini harbour does not seem right by a village, so you should expect to lose time going by bus or car to the harbour if you decide to book your tickets while on the island.
To summarise, as large ferries use this itinerary, if you are a pedestrian, you can still take the risk of booking last-minute and you will have a spot. If you have a car, you take more risks.
But anyway it depends mostly on you and on how flexible you are: if you really cannot plan ahead what schedule you would want, then wait for the last minute with the risk of staying one more day. If you have a tight schedule and prefer peace of mind, book ahead.
Regarding where to book, I was not inspired by websites to buy tickets, but neither was I inspired by the travel agents in the harbour. I think there is no better choice, but if you are concerned about the security of buying online, this is totally understandable and a good point.