TLDR: Yes, but not right now.
The case, on its merits
Is the HOA entitled to have a rule against inoperable vehicles in plain view? You bet it is. That's legit.
Does the HOA get to use reasonable, prudent and lawful methods to enforce that rule? Yes. The fact that the contract specifies no enforcement mechanism does not leave the HOA weaponless. The HOA certainly has a right to send a letter to that member - or all members - reminding them of the "no inoperative cars in plain view" rule.
The HOA can be a jerk about it, but so can the homeowner right back. This can very quickly turn into a "pissing contest" (is what they call it around here), where the homeowner responds with equal passive-aggression, CC's the letter to every other member, gets the TV news out there, puts political signage on the property (good luck getting that suppressed), and spoils the re-election chances of that board member and maybe others.
Remember, members in a society (e.g. HOA) have an absolute right to communicate with each other about the affairs of the society. Under the stress of a bitter fight, some boards think it's OK to regulate or impede that free communication. That is a strategic blunder that members can use to their advantage in all sorts of ways.
The HOA also has a right to engage in prudent and ordinary surveillence of the vehicle to ascertain compliance with that rule. Generally what the town does to detect the same thing - reading odometers through the windshield, marking a tire tread with chalk - is probably legit. I also point out the HOA is probably inside a city or township, and that may have laws pertaining to inoperative vehicles. The HOA does not override city law; that applies too. The HOA is within its rights to third-party the problem to the city, report the matter to the police, and let them deal with it.
That has the nice feature of involving a neutral third-party; so it isn't a tawdry Captain Ahab/Javert crusade of one HOA board member against some hated rival.
I suspect the police would immediately apply the tolling mentioned earlier, and say "We're not doing enforcement on non-moving vehicles, because parking your vehicle for 2 months is perfectly reasonable under shelter-in-place orders or advice for many vulnerable parties. People have enough problems right now."
HOA proceeding on its own
However, if the HOA decides to proceed on its own, they do indeed get to send the letter. This would almost certainly result in the car being moved from that spot to the next spot over, or reversed in the spot, or swapped with a car currently in the garage. This prima facie evidence of operability must end the inquiry.
If that does not happen, then the HOA should probably send a couple more letters as a CYA measure (just so the defendant can't raise a defense that "they were never made aware"). Then they would file a civil tort: breach of contract (again under normal circumstances; this isn't going to happen now). The defendant will consult a lawyer, who will say "just move the flippin car" to provide the above prima-facie evidence, at which point the HOA is obliged to drop the suit.
If the defendant is that stubborn (the lawyer may threaten to quit at that point), then the HOA will begin "discovery" or fact-finding. They will ask the defendant directly if the car runs, and the defendant is compelled to answer truthfully. (but doesn't the defendant have the right not to self-incriminate? No, that only applies to criminal charges.)
At some point, the defendant will realize "the jig is up" and deal with the car.
If pushed all the way to the end, the defendant will have to sit in the witness stand and answer questions in front of judge, counsel, jury (if opted) and spectators. The judge will say "Yeah, you have to answer that". And there will be a judge's order to move the car.
If the defendant loses utterly, and has no leg to stand on, the the HOA might ask for legal fees. But the award may fall far short of actual costs. And the defendant may not pay (the HOA can always attach the house with a lien, but that only cashes out when the owner sells. This can put the HOA dangerously cash-short.)
Then, the other shoe drops. The HOA has to raise fees by $200 to pay for the hole in the budget. Now that's being flipped against the HOA board: "those members bankrupted the HOA with cockamamie legal adventurism!" And every member has the $200 bill to remind them. So yeah, that board does not get reelected!
This won't even get docketed anytime soon
First, right now, that case isn't going anywhere.
I like to jump straight to the endgame and see what's going to happen if this gets in front of a judge. It won't.
Your state's shelter-in-place order just ended, but municipalities are at discretion to emplace their own, and in particular, individuals especially vulnerable people have every right to self-impose one. In fact, many people are being asked/advised to self-quarantine. This thing is not over.
Meanwhile, court activity has been greatly reduced, which means courts have a significant backlog of normal cases. And on top of that, they have armies of people a) suing each other over coronavirus itself. And on top of that, they have people suing each other over issues of contract performance that were realistically impacted by Coronavirus. Which your case falls into the category of; since Coronavirus all by itself constitutes a reasonable reason for them to not move their vehicle for an extended period.
Even more than that, look at your governor's emergency orders and/or legislation done for this purpose. Many states are tolling timing requirements, meaning whatever time limits are in place, the clock stops on those until the end of the emergency, which is still "on" in Kansas. Even if there was a rule of "move your car every 7 days", the rule is now 7 days past the end of the emergency typically.
In other words, the council member is being a huge jackass for choosing this particular segment of time to observe the car not having been moved. And is quite wrong to infer that therefore the car has broken down. Any judge will immediately recognize that!
Anyway, as said, court clerks are overwhelmed with cases. Court capacity is still diminished. So the courts will be slow to docket cases that a) as said, appear to be swayed heavily by Coronavirus, b) which appear to be related to timing issues affected by tolling, and c) which will probably sort themselves out by the time the thing realistically would get to court.
This case dings all those, so it's not going to get a slot on the docket. Period. If they even accept the filing, it will be deferred and deferred. And even if they got to the bench, parties will just get a lecture to sort it out.