Can I Implement a Well-Trusted Protocol with Well-Trusted Libraries in JavaScript securely? Just for anyone reading this: DO NOT ROLL YOUR OWN ALGORITHM.

Besides that, is implementing a well-trusted protocol with well-trusted algorithms and libraries a good idea? Is this likely to colossally fail like rolling your own crypto? Could this be used in production after enough peer review? (I know that rolling your own crypto is definitely NOT secure)

Let's say I'm trying to implement a simple JavaScript website that generates RSA key pairs, RSA encrypts, and signs, and also AES encrypts data (e.g. an asymmetric key exchange than a symmetric conversation; the simplest protocol out there). If I use well-respected libraries, respected algorithms (i.e. AES & RSA) and peer-review the website, is this likely to have vulnerabilities?

(We're ignoring all the issues of authenticity in code with JavaScript) (Also, I'm not implementing my own algorithm, i.e. AES, to keep that clear)
If you have to implement or design and implement your own protocol then using well established libraries is generally the way to do it. JavaScript is a large eco-system. For such ecosystems it is not adviced to implement your own (partial) "hazardous materials" library for the low level functionality that you require. It is much more likely that there is something out there that would be more secure.

Of course, if there already is a higher level protocol or protocol implementation out there then you should definitely try and use or improve that. Developing transport security - which is often what is tried with JavaScript - is very hard.  This is why most of the time the best solution is to use TLS underneath.