Will my plan be upgraded automatically once I use up my included returns quota?

No. The way Shopify has built their Billing for Apps is that only the Store Owner can approve a new subscription.

After you have exhausted your included monthly quota additional returns are billed per return ("Usage Charge") until the Usage-Charge Limit of your Shopify Account is used up. A notification in the Rich Returns' Dashboard will provide you more information on this . In your Rich Returns' Dashboard navigate to Account / Your Plan / and check your current return usage quota. The dashboard will also tell you what additional returns are billed at.