I'm not a mentor, but I'm in a similar situation. I have a math degree (with some CS) from a unremarkable school, no work experience, no network, and no web presence. I've had success getting interviews at large tech companies just by applying online. There's usually a job listing aimed at recent graduates. I should mention that I have fairly credible signals of CS skill on my resume (meaning I list some things I've built and say a few words about each), a decent academic record, and a fortunate aptitude for the sort of algorithm questions they like to ask during screening phone interviews.

To be clear, my first onsite interview is coming up, so who knows how it will turn out :) Be careful who you take advice from! Although this is very limited information, it seems I was under-confident that these companies would talk at all to someone who had e.g. no work experience and no referral. I would have told my past self to apply earlier!