Ultimately, this can be an accessibility issue if you want to deal with that.
This question on golf carts on trails speaks to other questions of trail user conflicts...bikes, e-bikes, skates, segways, etc. versus pedestrians. The greater mass of a cart adds additional hazard to pedestrians and cyclists. Not having lights also compounds the concern at night.
Given this is a small community, there might be a practical solution of allowing golf carts for those with a clear need, if there is a compelling reason, with clearly delineated rules such as carts yield to bikes and both to pedestrians. A very low speed limit for carts, and designation of the path as a shared-use cart path should be posted. This should be approached with great caution. A cart is really a low speed electric car that an unlicensed driver can operate. Safety issues should be considered.
Does allowing carts on a bike path open liability concerns? There may be value in looking at other communities with this sharing question for precedent and accident information.
Maybe a solution is to add a parallel crushed gravel adjacent pedestrian path.
Finally, who and why do folks need to travel by cart versus walking or biking unless disabled (at any age) or needing to haul loads like groceries. Maybe only folks with a disabled sticker on the cart are permitted and other solutions to haul groceries by handcart or bike trailer are implemented.
Ultimately, this can be an accessibility issue if you want to deal with that. Legally, you MUST allow people who say they have a disability to use golf carts/segways, or even ATVS unless you write a clear regulation for the trail to the contrary, and justify why some vehicles aren't allowed. If you wanted to limit it to people with a mobility issues you could use this to allow carts.
The Department of Justice specifies other power-driven mobility devices that can be used on trails by individuals with mobility disabilities. If you manage a trail on non-Federal land that is open to the public, this rule applies to your facility. This DOJ rule only applies to facilities covered under the Americans with Disabilities Act (ADA). Federal Lands are covered under the ABA, and not affected by the DOJ OPDMD rule.
American Trails has several related articles on power-driven mobility devices (see related documents).