Disagree with all these posts. If your documents state NO SOLICITING, the President or "any" homeowner in the community has the legal right to have them sent off the property.
At both entrances to our community are signs highlighted in red, and white.

rivate Property-No Soliciting:
It means just that. Soliciting is not just knocking on one's door.
You say your streets are private, so they must belong to your Association. Your Association if you are incoporated gives every shareholder the authority to take such actions.
If you are not going to stop ice cream vendors, then you cannot prevent soliticitations of any kind. If anyone is selling, or distrubiting information on resturants, pizza establishments, or a business of any kind in an effort to have you BUY something, then it is "clearly" soliciting.
A homeowner who invest their monies to buy a home governed by an HOA should expect some privacy.
Having to listen to the repeated music of an ice cream truck every day is not something one would describe as private.
Take the kids to the ice cream store, and if the majority of homeowners in your community don't have children
(I Do) I bet they would appreciate not having to listen to the annoying music, more then your kids enjoy the ice cream.
The old ruling is not to buy in a community having certain restrictions, and think ones you move in, you can change things.
If it is in the covenants, I agree with the President of the Board who is under fiducary guidelines to upheld the rules.
Probably get a lot of disagreements, but I intentionally purchased our home in an HOA community, and expect the Board of Directors, and the Management Company to enforce the rules I agreed to follow when I signed the line for my mortgage.
Thanks:
Jim