You can authenticate with the developer API via an API key. You can generate an API key from the Discord bot or via this API.
In order to access your teams data, you'll need your team ID. This can be put into the X-TeamId header, which tells the API which team you intend to manage.
X-TeamId:pfh0haxfpzowht3oi213cqos
You'll then need to put your API key into the X-ApiKey header:
X-ApiKey:{api_key}
Management Key
The management key can be used to access the developer API as well. This key will allow access to any team provided in the X-TeamId header.
Like-wise with the management API, you'll need to provide the key via the X-MgmtKey header.