What order should authors be on arXiv.org for equal contribution authorships?
If everyone made an equal contribution, you can put that information in the footnote. 
In a very famous (in the AI-related research) paper ADAM: a method for stochastic optimisation you can encounter the following line:


  ∗Equal contribution. Author ordering determined by coin flip over a Google Hangout.


I'd say randomisation is a fair way to go.
If you want to be more specific, here you can check examples for contribution statements.
There's a whole taxonomy for that: CRediT author contribution statements.
You can place these contributions e.g. just before the references section in the paper.

As Jukka already mentioned in the comment, arXiv does not matter in this case. 
I think the order should always reflect the real contribution (if possible).
All in all, it's good that you care about such things - good luck with your research.