There was a lot of work with sounding rockets after World War II. Those are rockets that go up and come back down--they are not intended to enter orbit, which is actually a much more difficult problem then just getting into space. For instance, Bristol Aerospace of Canada has a line of Black Brant rockets, like the Black Brant IV which can reach an altitude of 1,000 km, compared with the orbit of the ISS at 400km.
Here is Wikipedia's list of Black Brant rockets.
I don't know much about sounding rockets, and I couldn't possibly do them justice. Everyone gets excited about orbit, suborbital flights don't get as much attention. But Bristol is connected to Microsat Launch Systems.
Microsat Launch Systems, later renamed International MicroSpace, was founded in 1989 by Peter Diamandis, who has been a compulsive space entrepreneur for his entire adult life. The goal was to provide cheap and frequent launches of small payloads to low earth orbit. For the technology he intended to partner with Bristol, and use solid components from Morton Thiokol. His initial estimate was a launch for as low as \$1 million. As the company studied the technology and customer needs that cost went to \$3.5, then \$4.5, then \$6 million, while shifting away from Bristol and more towards Thiokol. The partner companies lost interest, investors lost interest, and the project sputtered out without ever building and testing hardware. I can't recommend any good web resources, but the history and context are very well explained in:
How to Make a Spaceship, by Julian Guthrie, 2016.
And the business history in:
The Orbital Express Project of Bristol Aerospace and MicroSat Launch Systems, Inc., but Geoffrey V. Hughes, 1996.
The first you may be able to find at your local library. The second would probably be an inter-library loan, but still obtainable.
The first private project was actually the Percheron rocket by Space Services Corporation of America, founded by David Hannah. The design, by Gary Hudson, was a simple pressure-fed kerosene and liquid oxygen design. It blew up during testing.
Then there was a series of Conestoga designs. The Conestoga I, in a 1982 test launch, was the first privately-funded rocket to reach space, but not orbit. The Conestoga 1620 would have reached orbit in 1995, but failed on launch.
Scott Manley has a good video on the Percheron and Conestoga story.
The first SUCCESSFUL privately funded launcher was the Pegasus rocket developed by Orbital Science Corporation. It has three solid-stages and an optional fourth hydrazine-fuelled stage, and is dropped at altitude from an aircraft. It was first launched in 1990 and is still in service today. A typical problem with funding private launch vehicles has been that investors haven't wanted to put money into a project unless there is a committed customer, and the customer doesn't want to commit unless funding and development is secured. The SDIO became the committed customer for the Pegasus, which gave investors some confidence that they would get a return. (Hughes, 1996, p. 2).
Then Kistler Aerospace was founded in 1993 by Walter Kistler and Bob Citron with the intention of developing a fully reusable rocket. It was to be a two-stage rocket, fuelled by kerosene and LOX. Unlike the dramatic powered landings that SpaceX is doing, the stages would have come down by parachute and landed on ground, cushioned by air bags. The development went way over budget, and NASA tried to keep it afloat in 2004 with awards that prompted protests from SpaceX because of a failure to compete the awards, as explained here. Scott Manley has a nice video on the Kistler story.
It is mentioned in the Space News link that Kistler was partially doomed by the failure of the telecommunications boom in the late 1990s, and the associated launches. That hit Boeing and Lockheed, too, as they were developing their Delta IV and Atlas V. The loss of the commercial business created a bitter rivalry for Air Force money which culminated in convictions for industrial espionage and, the creation of a joint venture, the United Launch Alliance, in 2005. Boing and Lockheed are traditional contractors, but in the 1990s they were operating as independent commercial launchers. I'm not sure where my other references on ULA are, but here is a video about the Atlas family by Scott Manley. Look in the reader comments for mattcolver1 and in the replies, Andre T, highly interesting stories from people who were there.
If the failures weren't mentioned there would be a lot less to say. Space is hard. Nobody seems to believe that until they try it themselves. There are exciting things happening today with technology, a business environment, and government policy making it the right time for it. It was hard to be a launch start-up before 2000.