Somewhere like Mt Buller can be an option - about three hours drive from Melbourne, it only requires snow chains under some conditions:
If you're driving to Mt Buller and staying overnight, you
will need to carry wheel chains by law. If you're visiting for the
day, you may or may not need to carry chains depending on snow and
road conditions. The daily wheel chain declaration is made each
afternoon and is communicated via a number of channels - click
here to view the current wheel chain declaration.
You can also catch a coach to and from Mt Buller without needing to drive at all and hence not needing chains. The times allow the majority of the day to be spent at the snow.
Mt Donna Buang is closer to Melbourne (less than 100 km away), frequently has snow, and doesn't necessarily need chains but are recommended:
Chains? Not usually needed but a good idea to have. No business in
Warburton supplies chains. As a gazetted road, VicRoads maintain and
grade snow from the roads and carparks - generally sufficient to get
vehicles without chains to the carparks. But if you have chains bring
them. Note there is no guarantee that 2WD vehicles will have access to
a mountain of snow. Chains and/ or 4WD are best - or seek advice
locally about conditions before heading up.