To build a Rest API that lets a client app (for example, a React APP) invoke Lambda functions via API Gateway, you should look at using the AWS CDK to create API Gateway endpoints.
Then you can use the AWS SDK for Java to implement Lambda functions that perform various AWS Service operations. For example, save data in an Amazon DynamoDB table.
Because there is a lot of detail involved in how to do this use case, this is a detailed end to end AWS App example. Consider this architecture illustration:

In this use case, the API Gateway REST API was created via the AWS CDK and the AWS SDK for Java v2 API was used to invoke AWS Service opertions and were implemented as Lambda functions.
To see the complete details, including the CDK Script (note that the link to the CDK script is located near end of the doc), see:
https://github.com/awsdocs/aws-doc-sdk-examples/tree/main/javav2/usecases/pam_source_files
If you spend time reading this and going through all the code, you will gain a much deeper undertanding how all of these pieces fit together. The final app looks like:
First it requires you to log in which authenticates with Cognito:

Then it invokes various AWS Service operations and displays a tag count of objects located in an S3 bucket:

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I'm trying to implement an app into AWS Lambda. It's currently one of those apps that uses @RestController and @PostMapping to create an API interface with multiple functions, but I'm trying to get it on Lambda. I know I must build a handler, but I'm not 100% sure how to do that.
Any help would be greatly appreciated.
EDIT: Going to clarify some details. The app is a Java application I was given, and it uses a Spring Boot Rest API. I need to covert it for API Gateway & Lambda, and I’m not sure how to do that, especially since the functions use specifically made JSONs, and I’m not 100% sure how to get it to work in this case.