Non accredited investors can invest in small private companies under SEC Rule 504. Stock that is bought is likely required to be held for one year or more. Also, there are severe limits to the funding amounts under the rule.
Otherwise there is an entire new set of rules for crowdfunding but the overall situation is similar to use of Rule 504 except that crowdfunding uses crowdfunding exchanges.
Most larger start-up companies require accredited investors.
Canada allows direct funding of operating companies by non-accredited investors. I can't particularly find a website that doesn't look junky.
More likely, significant development-stage companies can be found on the stock market. Judge these companies by cash on the books and not by touts.
Mutual funds have begun buying stock in large companies before the IPO. But the mutual fund is likely highly diversified.
A private-equity company might be found on the stock market that also participates in venture-capital.