NLP and one-class classifier building
Yes, this is feasible.
One-class classification is a thing, but it is usually used in a context where it is hard or impossible to get negative samples. In your case, I would argue, you can quite easily get tweets that are not about activism, therefore you can render it as a binary classification, because you have data points of two classes or labels: 1 for tweets that are part of your class and another 1 for tweets that are not. 
There are many ways to build a classifier, SVM is only one of them. You could also use a Naive Bayes algorithm, or as @Kasra mentioned a neural network model.
No matter what you use, you will have to organise your data such that you have samples of both classes: activism and non-activism within your set. This means that you should randomly pick tweets from your big dataset and manually check if they relate to activism, even if they don't have the hashtags in them that you used for identifying the activism tweets in the beginning.
Further, you have to think about the features that your classifier will use. The simplest might be the bag of words within the tweets, but you might also pre-process the tweet to exclude stop-words.
Depending on which algorithm you use, you might find that your classifier relies a lot on the presence of your particular hashtags as features for predicting the class. In this case it might struggle to identify other tweets without this hashtags as activism, even if they are activism. I would experiment with pre-processing the tweets in your entire dataset to remove those hashtags from the the tweets.