What are some techniques that I can use for anomaly detection given a non-Normal distribution? I have less than twenty available observations.
I would suggest a nearest neighbors approach. This technique is non-parametric, such that it does not assume your features follow any given distribution. The degree from which a novel instance can be classified as anomalous can set through some p-value estimation. These techniques are computationally expensive however due to your small dataset this may be well suited.
Learning Minimum Volume Sets http://www.stat.rice.edu/~cscott/pubs/minvol06jmlr.pdf
Anomaly Detection with Score functions based on Nearest Neighbor Graphs https://arxiv.org/abs/0910.5461
New statistic in P-value estimation for anomaly detection http://ieeexplore.ieee.org/document/6319713/
You can also use more rudimentary anomaly detection techniques such as a generalized likelihood ratio test. But, this is kind of old-school.