Yes, that is possible, a heat rash or simple allergic reaction.
But, non-blanching spots need to be examined in person, as they may be a sign of many dangerous conditions including meningococcal meningitis. This is not to scare you but alert you to concerning DD which needs to be ruled out.
Generally, other symptoms like headache and neck rigidity would be seen in the case of meningitis.
The rash progresses from petechial to a purpuric rash, which looks more like bruising than pinpricks.
It is best to see your doctor and get simple blood tests to rule out any concerning reason for this rash.
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