Which places I can visit near vaishno devi during december?

You are visiting in winter season. It's cold and many times snows in Vaishnodevi area,too. In case there is snowfall at the time of your visit, it will take more time than you must have calculated and all your travel plans may go awry. Please DO NOT plan many outings or far-off places. Jammu city is an interesting option. Various old and beautiful temples ( Raghunath temple, Ranbireshwar temple, Mahamaya temple and various others - a new on chenab canal is worth). Palaces - Mubarak Mandi and Bahu fort. Bahu gardens in Bahu fort is worth a visit. Then ofcourse lots of place for shopping ( dry fruits, kashmir silks and kaleens etc). You can travel further from Vaishnodevi towards Patnitop. Patnitop , Batote, Kud are few options. people stay in Patnitop ( it's costly ). You can opt to stay in Kud, if possible. Patnitop has few old temples ( sudhmahadev, gauri kund and nag temple). You can enjoy nice parothas just opp the entrance of Patnitop areas. While goint to Katra ( Abt 8 kms short of Udhampur, there is an ancient complex of temples at Krimchi. Worth a visit) You can opt to visit Akhnoor , few kilometers from Jammu city. A beautiful shiva temple while entering Akhnoor. Chenab river and Akhnoor- Chenab is famous for Sohni-Mahiwal love story. You can find ruins of Old Indus valley civilisation here. Please offer my prayers to Matadi. And my best wishes for your tour.

Jammu is one.