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According to Wiktionary, prodigo is a verb which etymology comes from "pro + ago". The same is suggested by L&S. However, I cannot see how ago fits here. The conjugation of this verb seems at odds with that of prodigo. Is there some mutation involved? How did proago become prodigo?

Before vowels the prefix pro- becomes prod-.
In addition to prodigere, we have prodire, prodesse, and maybe others that I forget now.
Where the -d- comes from is another question, but for practical purposes of learning Latin one can just learn the prevocalic version of the prefix.

It is very common that a short a in a short first syllable of a verb becomes i when a prefix is added.
For example, consider ex-facere > efficere.
This has happened with agere, too, turning it into -igere after (one-syllable) prefixes.

Once you change the short a into a short i in the present stem (otherwise the vowel or syllable is long) and remember to put the extra d in pro-, the conjugations of agere and prodigere match perfectly.

Joonas's answer is entirely correct, but just to add onto it a bit:

The way I learned it, prōd is an archaic form of prō. You'll also sometimes see pōr as in pōrrigō, pōlluō, etc: these three all came from different forms of the same PIE root.

In the end, prō was the one that won out, and so that's the only one you'll see used as a preposition. But all three of them show up in verb prefixes. Prōd was generally used in front of vowels, because Latin doesn't like vowels in hiatus (next to each other with nothing in between). And pōr was generally used before l or r, where prō would be awkward (forms like *prōrigō are somewhat unwieldy).

(It's also possible that prōd turned into prō, because Latin generally got rid of final ds: that's how we got filiā < filiād and mē < med (mēd?). But that's just conjecture; I don't actually know where the different forms came from.)

Re the vowel, Old Latin used to stress the first syllable of a word. And during this time, short vowels in non-final unstressed syllables (that is, any syllable besides the first and the last) got "reduced". The following rules aren't watertight, but they're decent guidelines:

• If the vowel was next to an r, it became e
  


• If the vowel came before two consonants, it became e.


• If the vowel was next to a "dark l" (short l not touching a front vowel), it became u
  


• Otherwise, it became i
  

You see this in a lot of prefixed verbs, like faciō ~ afficiō ~ affectus, or celō ~ occultus. The a was the original vowel here, but it only survives when it's in the first syllable of the word. (You also see this reduction a lot in reduplicated perfects, like cadō ~ cecidī. Basically any time you have a prefix before a short vowel.)

And that's what happened here: the original vowel was an a, but when it was pushed out of the first syllable, it got reduced to an i.

Once you make that change, you'll see that the conjugation actually lines up quite nicely! It's just one of the quirks of Latin that you have to get used to.