Recommender/Clustering data to support a hypothesis. Is this a valid use-case for unsupervised ML?
Clustering and recommendation are similar tasks, however in recommendation you usually want to recommend several items while clustering usually assigns each sample to only one cluster.
Anyway for your problem a clustering or even a classifier might help.
If labels are assigned on the basis of a similarity metric (and you have a good guess of what this metric might be) you could use a clustering algorithm to impute the missing labels (clustering is based on similarity between samples). However if it's reasonable that samples can be similar according to different metrics (and you are uncertain about it) then a classifier might help - provided you have enough labeled data to train the classifier. Which approach works best will also depend on the number of labeled samples and number of features.
Whichever strategy you choose to use the imputed labels will be noisy (some samples will be mislabeled). This is important to take into account especially if you are going to further process those labels.