biblatex is not a problem for you. biblatex is just a normal LaTeX package and that should work independent of the system. Maybe the OpenBSD people did not want to include it because Biber is missing, maybe it is available somewhere. But you can always install a vanilla TeX live instead of the often outdated repository version of TeX live. Or (shudder) you can install the package manually.
Your real problem is Biber. Biber is biblatex's replacement for the BibTeX backend. Like BibTeX Biber is an external program (Question mark or bold citation key instead of citation number explains what the two do).
Biber is a Perl program and precompiled binaries are available for several systems via TeX live (and MikTeX), the binaries can also be downloaded from SourceForge (https://sourceforge.net/projects/biblatex-biber/files/biblatex-biber/current/binaries/).
Unfortunately, there are no binaries for OpenBSD. Instead you can download the Perl sources for Biber (https://sourceforge.net/projects/biblatex-biber/files/biblatex-biber/current/) and run Biber as Perl program directly. This should work if you have Perl 5.26 or above and can install all required modules for Biber. See also §4 of the Biber documentation.
If you can't get Biber to work on your system you can consider using backend=bibtex and running BibTeX instead of Biber. In general it is recommended to use Biber with biblatex, and not all features will work if you use BibTeX, but this could work for you as a last resort.
Depending on your document it might not be too hard to go back to natbib, but if you use the cool features of biblatex you might find that not everything can be done that easily with natbib. \parencite and \textcite are your smallest worry: Change Biblatex commands to Natbib.
Going back to Linux would eradicate the problem. But I assume you had a reason to switch to OpenBSD.
biblatex.biblatexis just a normal TeX package. Your problem here is that Biber is not available for OpenBSD (there are FreeBSD versions - I have no clue about the BSD style systems - is there a chance that binary could work for you?). You may get away with using the Perl version directly (I have never tried that), you can get everything from https://sourceforge.net/projects/biblatex-biber/files/biblatex-biber/current/. If you don't need the Biber features you can try using onlybackend=bibtex. – moewe Apr 25 '18 at 13:35