Yes, Alexis left on her own accord when Erynn and Ethan started fighting, no instructions from Erynn or anything else.
Did she set out to look for the MC?
We can't make any positive affirmations, but she doesn't run after the MC right after he runs away, she leaves non-chalantly after her friends start fighting with each other(it's hard to judge passage of time in something like this, but there has been quite the back and forth between Erynn and Ethan already... 2 min maybe?), to me she ran into him by chance and seeing him sad decided to say something nice, she wasn't going into his house to speak with him like Erynn tried later.
As the scene is portrayed, her main impulse for leaving is that Erynn and Ethan fighting is not something she enjoys being around, not thinking that the MC might be feeling bad and needs a friend right now.