Loyal3(
https://www.loyal3.com/) periodically gets some shares allocated during an IPO, that the general public can buy via their platform. GoPro and Square are a couple of their notable offerings.
Another option is the pre-IPO market. EquityZen(https://equityzen.com/) is one of the secondary market where you can buy shares pre-IPO from other shareholders. (Disclosure: I am an engineer at EquityZen)
