We should start with distinction of project management as role and project manager as job. Some tasks which are typically understood as project managerish are there even in small projects, however they don't have to be done by a person who has "project manager" printed on a business card.
Usually, as the organization grows, first people who start acting like project managers are team managers. After some time company leaders realize it would be beneficial to have project managers and let team managers do their job, which is leading teams. If I had to guess I'd say it happens when company size is somewhere between 25 and 75 people.
However there are a lot of specific situations where it would work differently. One of examples can be agile organizations, where formal project manager job doesn't have to be introduced as their tasks are spread over different roles, e.g. in Scrum over the whole team, including Product Owner and Scrum Master. If the company follow this kind of approach rather conservatively they can avoid hiring project managers for quite a long time, which of course doesn't mean project management tasks aren't done.
Also it is usually important what kind of clients you deal with. The bigger and more formalized organization as a client the faster you have project managers on your side as those clients just expect that you have them in your team.