There are some online algorithms for SVMs do exist, so it is important to specify if you want kernel or linear SVM, as many efficient algorithms have been developed for the special case of linear SVMs.
For the linear classification case, you should use the SGD classifier in scikit-learn with the hinge loss and L2 regularization, then simply you will get an SVM that can be updated online. You can combine this with feature transforms that approximate a kernel to get similar to an online kernel SVM.
You can use the PassiveAggresive classifier that will give you better results, as it's learning rate does not decrease w.r.t. time.
While training you can attempt to detect decreases in accuracy over time and begin training a new model when the accuracy starts to decrease (and switch to the new one when you believe that it has become more accurate). It also has more online linear and kernel methods.
Hope this answer helps.
Learn SVM with this Support Vector Machine Tutorial.