Can a private company issue digital share certificates?

The company is private and registered in Quebec. I would like to sell low-value shares internationally. To make small investments feasible, we'd like to not have to print out physical certificates and mail them. Could we act as our own transfer-agent and say that our shares are "book entry shares" ? In the ideal scenario, we wouldn't have to send a physical certificate, and what we'd send electronically would serve simply as a receipt.

As much as I understood, a private company is not one owned by a single person, but one that is not publicly traded. The hypothetical situation I'm asking about involves private offers over the web.

I am confused by your terminology. You say that you are a privately held company, which, by definition, means that you own the whole company and there are no shares. In order to have shares to sell you would need to convert to a publicly held company.

You can not sell "internationally". The transactions would take place in some country and you would have to register and comply with the laws of that country's regulator. If you want to sell in many countries, you would need to comply with each country's rules, one country at a time. In the US, the SEC does allow companies to do away with paper certificates. In Britain, I don't know if the FSA does or not. Each country may require a local registrar and transfer agent, or not. So you may or may not do this yourself, depending on the country.

No.