Blimey... can I do it in Type 10?
You can do the hauling in any ship that can carry cargo, and you don't specifically need an Alioth permit. That's just the closest system that was selling the commodities you need for this CG, and it's pretty much sold out anyway.
Two important things:
1) CG will run until some time in the early hours on Thursday, so effectively it finishes Wednesday night for UK-based players.
2) To be eligible for the CG rewards, YOU HAVE TO SIGN UP TO THE COMMUNITY GOAL BEFORE YOU HAND IN ANY CARGO! You can sign up at any station with a mission board, just scroll to the top, and pick the CG that's based in Alcor (not the one in Colonia). That said, you can sell cargo without being signed up and still get the credits, so it's not a disaster if you forget, you just won't have a chance at the engineered modules.
At this stage of the CG, most of the commodities near Alcor have sold out, so you want a ship with decent jump range - that has to be balanced against your cargo space, of course. Chances are, your Conda might be the best suited to the job, if you refit it for cargo hauling and lose any unnecessary weight.
You can use eddb.io to search for stations selling the three commodities in this CG. There used to be a way to permalink a search on EDDB, but I can't find it so I guess it's been removed. The three commodities are:
https://eddb.io/commodity/64 (Auto-Fabricators - lowest profit, so more availability)
https://eddb.io/commodity/300 (Micro Controllers - second-best profit)
https://eddb.io/commodity/69 (Resonating Separators - best profit, lowest availability)
In the search options, enter "Alcor" for the Current System, and select "No" for Include Planetary/Odyssey/Fleet Carriers (landing on planets takes ages, and Fleet Carriers are often not accessible to everyone). Choose a landing pad size appropriate for your ship, and I recommend searching for stations with at least 10,000 Min Supply, or you'll be racing to get there before it all sells out. Choose "Buy" for the "Buy or Sell" dropdown.
Once you have your results, you want to prioritise stations which are close to Alcor, have RECENT update times (ignore anything that's days, or even hours, old), and which aren't too far from the entry star (that's the column with numbers followed by "ls").
Good luck! Or just have a lie down and remind yourself that credits aren't real, and there's always a next time