Placing an AWS Lambda in a public subnet
Provided the Public Subnet where you're launching your Lambda is configured with Auto-assign public IPv4 address: Yes it should work in public a subnet without NAT.

Whether or not it's a good idea is another question. In general you shouldn't launch any backend services in the public subnets, the public subnets are for things like API Gateways, NAT Gateways, Load Balancers, etc. Not for launching the actual backend systems. That's the best security practice. 

However that aside I'm pretty confident that with auto-assign IPv4 address enabled Lambdas will work even in the public subnet.

Unless you want the Lambda to connect to any VPC-internal resources (e.g. to your RDS) you may just as well launch it without VPC config (leave those VPC and Subnet settings empty). That will give it access to the internet without having to worry about NAT or public IPs.

Hope that helps