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I want to create a Lightning Web Component (LWC) that be included in a managed package. Our current process uses SublimeText and Mavensmate (which I know is deprecated). We also use GitHub as our repository to move code and it is deployed using CircleCI. We don't want to re-do our entire set up because this LWC is only a small part of the larger package.

However, every tutorial I've seen for LWC uses SalesforceDX to set it up. Is there a way to create a LWC without SalesforceDX? I understand that Mavensmate doesn't support LWCs but are there other options? I'm even willing to use the Developer Console if needed.

You don't need SFDX at all (yes, even Salesforce CLI is not needed). It's fully supported by Metadata API.

Personally I'm using Illuminated Cloud plugin for deploying/retrieving LWCs but it's also possible to use Ant Migration Tool to do that (or other tool that's currently supporting deployments at API 45)

Add to yours package.xml:

<types>
 <members>*</members>
 <name>LightningComponentBundle</name>
</types>

Its available since API 45.

There is one limitation though as @Jayant Das mentioned. You cannot use Developer Console to create them.

Docs

UPDATE:
Updated the answer to reflect Metadata API support for LWC.

You don't need the full fledged Salesforce DX here. To be able to use LWC, you can use the following.

• 
  Salesforce CLI. To be able to create projects, move components from sandbox to sandbox


• 
  Metadata API. If not using CLI, you can use this to move components from sandbox to sandbox


• A code editor (preferably VS Code). You cannot use Developer Console to develop LWCs.

The following documentation provide more details on this topic:

• 
  Set Up Your Development Environment


• Development Workflows


• 
  LightningComponentBundle Metadata API

I personally use VSCode. It is very easy to setup if you follow these steps to install it with all of the salesforce supported extensions https://forcedotcom.github.io/salesforcedx-vscode/articles/getting-started/install. Then, link your ORG(NOT scratch org, your sandbox or prod org) following these steps https://forcedotcom.github.io/salesforcedx-vscode/articles/getting-started/install
I hope this helps.