Is it possible to still implement a program that modifies the audio frequencies (but only relying on the DSP of the host device i.e Apple, Android device), and then transmits the modified frequencies to the speaker?
I can give you an absolute, positive, definite maybe.
You absolutely positively can do digital signal processing in your cell phone's CPU -- digital signal processing is just math, and can be done in any general-purpose processor. The only thing special about a digital signal processor is that it's heavily optimized for the sort of computation that's needed for most DSP work.
The presence of dedicated DSP chips on the market is fading these days, because general purpose processors are getting cheaper and faster in general, and because general purpose processors are getting more and more "DSP-ish" features. So there's a good chance that a new device will have plenty of raw horsepower.
Where the "maybe" comes in is whether the phone's operating system supports doing real-time DSP on the processor, whether the processor has sufficient computational power do implement your desired algorithm, and how easy the phone's operating system makes it for an app to do the work. The answer to the OS part of that "maybe" is a question for Stack Overflow or some other more "pure" software-engineering oriented site; the answer to the processor loading part of that "maybe" is for you to determine experimentally once you've figured out how to write an app that can diddle with the audio stream.