Learn Finnish: Onko uusi sovellus kehitteillä vai näytteillä? - Is the new application under development or on display?

They are formed from the nouns kehite (a development or draft) and näyte (a sample). These nouns are put into the plural adessive case (-illä). This structure is commonly used to express that something is in an ongoing state or process.

The lesson mentions these forms often take a possessive suffix. Why are there no possessive suffixes on the words in this sentence?

Possessive suffixes are added to these adverbials when the action is tied to a specific actor's ongoing process (for example, Hänellä on kirja tekeillään, "He has a book underway"). In this sentence, kehitteillä and näytteillä simply describe the general, passive state of the application itself, so no possessive suffix is needed.

What are the root verbs for these process adverbials?

Kehitteillä comes from kehittää (to develop), and näytteillä comes from näyttää (to show). You will often see this pattern where a verb is turned into a noun ending in -e, and then put into the plural adessive to describe an ongoing state.

Could I use tai instead of vai here to mean "or"?

No. In direct questions where you are offering two mutually exclusive alternatives (it is either one or the other), you must use vai. You use tai in statements or when the options are open-ended.