You can create your own VPC and use it for MWAA. There's nothing special about the one created by default. It is a CloudFormation stack that makes a VPC, some subnets, security groups, routing tables, etc. Your VPC will simply have to provide equivalents of these. It will be more work to create and maintain your own VPC.
However, you say that you want to do this because you are trying to access an RDS from Airflow. It is irrelevant what VPC the target resource is. Airflow can connect to other VPCs, other AWS accounts, even other platforms like Azure or GCP. Remember that Airflow itself is not AWS specific, it is a generic open source program that can be installed on any type of server.
You will simply have to configure an AWS connection for the RDS you want to access, and put in things like its URL, username and password there. Presumably you can connect to this RDS from your own computer. Airflow can do the same. You want to check the access settings of the RDS, in case there's something like blocking all IPs by default but allowing your PC's IP through - you'll obviously have to add Airflow's IP to the exceptions in that case.