Deriving a word for the activity of using a tool from the tool name ("grep")

This isn't a very technical answer, but googling "grepping" returns 354,000 results. Googling "greping" only returns 47,300 results and suggests that you meant "grepping" instead. It seems that "grepping" is the correct usage.

Since grep (pronunciation) rhymes with step, I would follow the pattern with stepping and write grepping.

(Writing it as greping makes me want to move short e (/ɡɹɛp/) in the first syllable to a long e, /ɡɹip/, rhyming with weeping. Your mileage may vary.)

Similarly, while awking seems straightforward, I'd favor sedding over seding.

Arguments in other answers for doubling the p are compelling, but also note that Wiktionary (linked to but not quoted in another answer) specifically shows such spelling of grep's present participle, among other forms:

grep (third-person singular simple present greps, present participle grepping, simple past and past participle grepped)

making it clear that at least one dictionary shows the p in grep doubling in its participles.

I was taught in elementary school that if a syllable ends with a vowel, the vowel is normally long, while if it ends with a consonant, the vowel is short. Also that if there is a vowel followed by one consonant in the middle of a word, the consonant is part of the next syllable, while if a vowel is followed by two consonants (that do not work together to make a single sound, like "th"), that the first consonant ends the first syllable and the second consonant begins the second syllable.

Thus: "hatter" - syllables are "hat" and "ter", "a" is short.

"hater" - syllables are "ha" and "ter", "a" is long.

(Yes, there are lots of exceptions, but I was taught this as the general rule.)

As a consequence of this, if a word ends with a single vowel followed by a single consonant, than to keep the vowel short, you must double the consonant when adding -ed or -ing.

Example: "fit" becomes "fitted" and "fitting".

There is no need to double a consonant if the word already ends in two consonants. Like "list" becomes simply "listed" and "listing".

So by these conventional rules, "grep" would become "grepped" and "grepping". If you wrote "greped", we would make the first "e" long, like "greep'd".

Hyphenated suffixes are usually an idiosyncratic choice that mainly serves stylistic purposes; as far as grammar is concerned, hyphens are normally reserved for words where the suffix is a proper noun or itself a large word (three or more syllables), although there is no consensus and there are exceptions to the rule.[1]

For me there is no need to put quote marks around it either. There is an established body of similarly-derived words in reputable dictionaries by now. That is, names of digital tools used as verbs.

Both Merriam-Webster & Oxford: photoshop, google
Oxford only: skype, even facebook(v.) !!

The only thing left to consider is whether to duplicate the last letter of the root. This has been tackled in other questions[2], although again there is no consensus and in my experience it is often a matter of dialect. It is a lot more common to omit the duplication in American English than in British, despite the obvious challenge that omission creates in pronouncing the word correctly.

In any case, Oxford Dictionary gives the word as grepping.

grep | verb (greps, grepping, grepped)

[with object]

Search for (a string of characters) using grep.

---

References:

[1] http://www.dailywritingtips.com/close-the-gap-on-prefixes-and-suffixes/

[2] What is the rule for duplicating the last letter when adding "-ed"?

Simply stick "-ing" to a noun to verb it:

• tool -> tooling
• shed -> shedding

The reason for the extra "d" in "shedding" is that the "shed" part of "sheding" sounds different.