How should a Play application be set up to handle requests sequentially at a time?

The simplest approach is probably, expanding on Feyyaz's answer, to define a single-thread ExecutionContext:

val singlethread = ExecutionContext.fromExecutorService(Executors.newSingleThreadExecutor())

I recommend not making this context implicit, as you want only the execution of the command to happen in that context.

val execution = Future("command_to_run".run())(singleThread)
execution.map(_.exitValue)  // will execute on the regular execution context in scope

The single thread context will then be in use for no longer than required to run the command and will then be available to serve the next request. In general, it's not a great idea to have

import scala.concurrent.ExecutionContext.Implicits.global

in production code, unless you really know what you're doing (IMO that the compiler suggests this is a horrible misfeature).

It's also worth considering using an Akka actor to synchronize running the command: this will simplify failure handling/cleanup and not require anything special in terms of ExecutionContexts, at the expense of needing to work with the ActorSystem Play supplies.