There has to be a shutoff valve somewhere inside your house or in the crawl space. It might not be located close to the green valve box. Perhaps try turning on the sprinkler, then go into the crawl space and listen to the pipes to see which one has water flowing? This might allow you to trace that pipe and hopefully find a shutoff.

Is there a garden hose spigot located nearby? The sprinkler and that spigot may share the same shutoff valve (mine does!). This also allows me to close the shutoff, open the spigot, and then use my air compressor to blow out the sprinkler lines. Unlike what the other commenter said, this is completely doable by a homeowner.

I use a 24 gallon air compressor and set it to about 70 psi. I cobbled together a short hose that has an air compressor quick-connect on one end, then a 1/4 turn shutoff valve, and then a garden hose connector on the other end. The process is simple: shut off water supply, open spigot, attach hose assembly, activate one sprinkler zone, then slowly open the 1/4 turn valve to pressurize the line. Given the size of my compressor and the long runs for each of my zones, I usually have to cycle through the zones a few times to ensure all water is out. Good luck!