Qualified yes, but it becomes more difficult (and depending on the features required impossible)
Devices with Google Play has a set of built in services - Google Play Services - which is provided for free (or nearly so) for apps. These services include Location & Maps, push messaging (was Google Cloud Messaging, now Firebase Cloud Messaging), Google Sign-in, and others. These services provide a lock-in to developers as they are available in any device that comes with Google Play Store.
All of these services are subject to change over their lifetime and Google has already announced that Google Play Services for Ice Cream Sandwich (ICS) API 14 & 15 will no longer be updated
What this means is that for any app developer interested in the current market of Android devices with Google Play Services, they will have to do additional work to support older devices where the Google APIs are frozen.
Another pain point for developers is that the WebView prior to Android 5.0 was tied to OS updates so WebView issues/quirks varied depending on manufacturer and OS version.
You would also need to find older web browsers as both Firefox and Chrome are at or above Android 4.0. So advances in mobile website development may break on older browsers.
As mentioned by @Robert:
The Android Tools Team has evolved the build tools from the time of Gingerbread: Eclipse IDE based to IntelliJ IDE, Java Ant build system to Gradle build system, etc. There may be a situation where either the tools no longer support the older SDK or the developers for old devices need a depreciated API which is no longer supported by the latest SDK or any recent ones. Additionally Android developers depend upon third-party support libraries written by Google or other developers each of which have their own depreciation schedule.
What can I do
Run a Custom ROM with a replacement for Google Services if necessary, see: Fakestore, blankstore, gms, microg, unifiednlp - what's best for Android without Gapps?
What if I don't want to root/Custom ROM
Any developer would need to find matching APIs for the Google Services which would work at the lower API of the device. They will probably need to pay for things which were free with Google Services like in-app map tiles and push notification.
Note that even the open-source app store F-Droid now has a minimum API level of 14, Ice Cream Sandwich
Developers are still free to create apps which users can download and side-load on older devices, if they don't use any deprecated Google Services for that device