Will anything happen to a marble which placed in vinegar solution?

No. A marble is made of glass, not marble. Glass will not react with vinegar.

Edit - Some antique marbles ARE crafted out of marble or other stone/ rock types. Using ANY sort of chemical to clean them can possibly damage the surface finish or even worse, cause the marble to break or split. I happen to own a few antique marbles from the 1880's to the early 1900's. To clean them use a damp, warm soft cloth. Try checking each marble by carefully rubbing a small portion of a marble before cleaning the entire marble(some marbles are sun hardened clay, and getting one wet can damage them).

One other method to identify the material your marble(s) are made out of - try a local rock shop. Most rock shop owners can identify what your marble is made from, if it is a naturally occurring rock. They might even be able to provide the possible value of your marble(s), but you should ALWAYS get a second opinion for any value appraisals.