Though it’s been suggested that running in the morning is a risk factor for running injuries, a literature review in 1985 established that wasn’t the case.
While this is a very old paper, the majority of the most recent research has focused on other risk factors such as volume of running, previous injury, etc. The reason they focus on these is because they’re the most common, which suggests that the risk factor of ‘early-morning running’ is lower down the potential-to-cause-injury list. That’s not to suggest it’s not a credible risk. One study talked about injuries while early-morning running due to slips and trips when running in the dark! But of course the other reason that early-morning running may be a risk factor is if you’re not properly warmed up.
As you sleep your body cools down, meaning that when you wake up your body is colder. This includes your muscles, ligaments and tendons – all things that might get injured if not warmed up properly. So if you get up, grab your shoes and go straight into a tempo run (not advised!) then you’re going to be more susceptible to injury.
I advise making enough time for a full warm-up before any run, and especially first thing in the morning.