Hi @Austin-3185
One way to achieve this that I know of would be to save the context of the conversation ( Bot Framework State ) to Azure Cosmos DB or Blob storage and have your API listen to the storage. You can chose which property bag you want to store, like UserData or ConversationData or use custom prompts to collect the data from the user.
This can give you an idea: https://learn.microsoft.com/en-us/azure/bot-service/bot-builder-howto-v4-storage?view=azure-bot-service-4.0
You can then have an Azure Cosmos DB or Blob Storage function trigger to run your api when there is a new entry in your storage: https://learn.microsoft.com/en-us/azure/azure-functions/functions-create-cosmos-db-triggered-function
Hope this helps!