It is possible to trigger azure devops release pipeline stage via rest api.
You can first use Releases-Create rest api to create a release.
POST https://vsrm.dev.azure.com/{organization}/{project}/_apis/release/releases?api-version=6.1-preview.8
If you want to trigger a specific stage in this release. You can then use Releases - Update Release Environment.
PATCH https://vsrm.dev.azure.com/{organization}/{project}/_apis/Release/releases/{releaseId}/environments/{environmentId}?api-version=6.1-preview.7
See below example in powershell script: You can follow the steps here to get a personal access token.
$url = "https://vsrm.dev.azure.com/{org}/{proj}/_apis/release/releases?api-version=6.1-preview.8"
$PAT="Personnal Access Token"
$base64AuthInfo= [System.Convert]::ToBase64String([System.Text.Encoding]::ASCII.GetBytes(":$($PAT)"))
$body = '{
"definitionId": releaseDefinitionId,
}'
#Create a release
$releaseInfo = Invoke-RestMethod -Uri $url -Headers @{Authorization = ("Basic {0}" -f $base64AuthInfo)} -Method post -Body $body -ContentType "application/json"
#Trigger a specific stage
$releaseId = $releaseInfo.id
$EnvironmentIds = $releaseInfo.environments | select id
#Trigger the second stage
$EnvUrl = "https://vsrm.dev.azure.com/{org}/{proj}/_apis/Release/releases/$($releaseId)/environments/$($EnvironmentIds[1].id)?api-version=6.1-preview.7"
$envBody='{
"status": "inProgress"
}'
Invoke-RestMethod -Uri $EnvUrl -Headers @{Authorization = ("Basic {0}" -f $base64AuthInfo)} -Method patch -Body $envBody -ContentType "application/json"