Will skipping Play Store updates cause problems?
Android applications are used to perform a task or a  group of tasks. App updates are additional features that are added to the current version of that app. An application has several states while running on mobile such as start, stop, resume, terminate etc. 

Play Store is one of the secured source for downloading and installing android applications. A developer in play store will release two types of app(not for every apps).


Stable :
This version runs stable on most devices and may not have any app close or similar issues.
Beta :
Releases updates under development stages and not upto stable. User need to join beta program from play store.


So, if you have installed an app from play store, then you can run that app without updating for long time, because that is stable. The only thing you've missing will be updated features.  For example, Google phone app(say version 23). It had added new design few months ago, but if you're running version 22, you can use as long as you can. And you'll only miss new design, dark theme like that. And, if you're installing version 29, after skipping 23,24,.. , you'll get features of that versions, after updating to version 29. It's because android apps are developed  incremental basis( as Firelord mentioned below).

Also, there are some apps which requires strict updates. Example, WhatsApp. WhatsApp will check the current version of app and prohibits app usage until app update. This occurs because WhatsApp may stop support specific version after few months(may be yearly, you're redirected to  play store while opening WhatsApp).

System apps may get auto updated during OTA/ security updates released by manufacturer.

If you're installing an app from play store today, you're installing today's latest stable version.