Can I preform the ANCOVA test without the assumption of normality? If not, what should I use instead?
You can use the rank analysis of covariance (RANCOVA) first described by Quade:
Article Rank Analysis of Covariance
See also:
https://digitalcommons.wayne.edu/cgi/viewcontent.cgi?referer=&httpsredir=1&article=1924&context=jmasm :
The RANCOVA of the ranked posttest on the ranked pretest ascovariate for the Group factorand the tests on the residual ranks all gave pvalues
ANCOVA is relatively robust to departures from normality, so I would ask just how badly are you violating the assumption. In many cases, you are safe thanks to the Central Limit Theorem. For a more detailed explanation, you can check out this video: https://youtu.be/1vMGC66yiqI
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