Please refer to Jörgen Bergström's blog about this: http://techstuff.bergstrom.nu/429-rate-limit-exceeded-in-logic-apps/
Essentially he says you can setup multiple API connections that do the same thing and then randomize the connection in the logic app code view to randomly use one of those connection which will eliminate the rate exceeding issue.
An example of this (I was using SQL connectors) is see below API connections I setup for my logic app. You can do the same with a blob storage connection and use a similar naming convention e.g. blob_1, blob_2, blob_3, ... and so on. You can create as many as you would like, I created 10 for mine:
You would then in your logic app code view replace all your current blob connections e.g.
@parameters('$connections')['blob']['connectionId']
Where "blob" is your current blob api connection with the following:
@parameters('$connections')[concat('blob_',rand(1,10))]['connectionId']
And then make sure to add all your "blob_" connections at the end of your code:
"blob_1": {
"connectionId": "/subscriptions/.../resourceGroups/.../providers/Microsoft.Web/connections/blob-1",
"connectionName": "blob-1",
"id": "/subscriptions/.../providers/Microsoft.Web/locations/.../managedApis/blob"
},
"blob_2": {
"connectionId": "/subscriptions/.../resourceGroups/.../providers/Microsoft.Web/connections/blob-2",
"connectionName": "blob-2",
"id": "/subscriptions/.../providers/Microsoft.Web/locations/.../managedApis/blob"
},
...
The logic app would then randomize which connection to use during the run eliminating the 429 rate limit error.