All the tutorials I have seen referencing the use of Visual Studio Code for Salesforce development imply that it is used in conjunction with SalesforceDX. It is my understanding DX is a relatively new development technique/workflow for SF… git, easier deployment, component based instead of whole org, etc. In my current environment, SalesforceDX has not been adopted.
Is it possible to use Visual Studio Code in that “traditional/older” development workflow without DX specifically? My assumption is that it will be similar to how Mavensmate and Sublime would have functioned.
There are some VisualStudio code extensions that can work without needing SFDX workflow. Note that current official plugin also supports working without DX scratch org support.
I created my own extension to supplement Salesforce official visualstudio extension pack; feel free to use that. Note that you will need Salesforce DX CLI and the plugins (Does not mean you need DX scratch org capabilities or hub org). It makes saving apex, vf and lightning components faster and works along side of the official extension.
ForceCode is another option.
There is also a mavensmate extension package for building Salesforce applications.
There are also bunch of few more extensions that you can try and see if it works for what you want.