Ok... interesting question! I don't remember reading/seeing that exactly in the context of the Revolutionary War. But... I can say that all the way up to WW1 in the UK and through the Spanish American War in USA, the 'private raising of regiments' was completely commonplace and these units or regiments would supply their own equipment.
Some well-known-examples? Well, Morgan's Riflemen was one. Daniel Morgan of the Revolutionary War organized his own unit that was critical to the battle of Saratoga. These folks were rifle shooters (a rarity in an era of muskets) and had to supply their own rifles, ball and powder. Which made sense, because to shoot a rifle accurately, everyone had their own special 'recipe.' Berdan's Sharpshooters were another famous regiment raised by an individual in the Civil War.
The American Civil War saw a lot of this. And if you raised, paid-for, and equipped your own regiment, you got to command it and choose your officers, etc. It was like little regional private armies. Fire Zouave's were marginally in this category. But more telling were officers like Benjamin Butler who raised his own regiment and showed up like a rash all over the Civil War. Not known for competence, but for his wealth and position as, IIRC, a Massachusetts politician. He bought, personally, 12 Gatling Guns for $1,000 each (a lot of money at the time) and equipped his regiment with them at Petersburg. He owed his position (and his not getting sacked) to his powerful political position and boatloads of money.... some of which is believed to have been stolen when he looted New Orleans (he was military commander there)... stealing so much silver that he became known as "spoons" Butler.
In fact, the situation in the Civil War got so bad with incompetent 'political generals and officers' who were rich enough to raise their own regiments that Grant had to dial lot of them back and fire generals from their own regiments.
Even in WW1, the first regiment of the famous Rolls-Royce armored cars was raised and equipped by the Duke of Westminster, who bought 12 Rolls-Royce chassis and had them turned into armoured cars. He took his squadron to France, German West Africa (now Namibia), Gallipoli and, later, to Egypt where they staged the world's first mechanized POW/hostage rescue (the Tara Incident) and worked with Lawrence of Arabia. The Duke was the wealthiest man in the UK at the time and he commanded and supplied his own troops. And got the medals to prove it. He was, in fact, quite competent!
So in the roundabout way of answering your question is that there is little doubt that wealthy private citizens in the Revolutionary War raised their own units and regiments and probably supplied, out of their own pockets, heavy weapons. At the time, foundries and weapons factories, other than the National Armory system, were a cash and carry business. If you wanted a cannon or three on your plantation, then you went to the Tredegar Iron Works, handed over your money... and there you go. And in the South, local militias were very dependent on the local plantations by the Civil War... I have no doubt that in pre-revolutionary war America, the frontier/western plantations and large land-holders undoubtedly had heavy weapons for their 'frontier defense.'
I would also be willing to bet that there were more than a few who dug deep in their pockets... and bought cannons and other things for either their local regiments... or simply bought them 'for the treasury.' Again, for historical reference, there were towns buying tanks and planes and Howitzers during WW2 in bond drives... raise $150,000 and your town can send a tank to the boys! Lots of Daddy Warbucks types did this, for sure.
So while I can't point you to a single source that says 'yes' for the pre-revolution America and the Revolutionary War, practice in our parent country was that private citizens supplied regiments... and the practice in our own country was the same in subsequent wars, right up through Teddy Rooselvelt's Roughriders and the Spanish American War. So odds are, that it was just as common c. 1776.
Hope that helps,
Cheers,
Sirhr (Resident Historian and artillarist...)