I would say that, rather than being a matter of either grammar or inflection, the use of -t- or -tt- is just a convention about the spelling of these words. The pronunciation of the double -tt- in "omitted" doesn't contrast with the pronunciation of the single -t- in a word like "literally" or "mitigation". The usual phonemic transcription of a word like "omitted" would only include one /t/. The inflectional suffix is the same in both vomiting and emitting: it is -ing.
As outlined in the answers to "Focussed" or "focused"? Rules for doubling the last consonant when adding -ed and various other questions on this site, the use of double consonant letters before certain vowel-initial suffixes in English spelling is very closely correlated with the stress pattern of a word (and also with the type of vowel sound preceding the consonant).
Many English verbs that have forms spelled with -tt- can in fact be traced back to etymological sources with phonetically long /tː/ (in Old English, in some other Germanic language, or in Latin), but I would say that the relationship is only indirect: we see this correlation because a vowel before a historically single -t- was often either unstressed, or lengthened if stressed. For example, the verb hate historically had a short /a/, but this was lengthened in Middle English. The set of lengthening changes like this presumably contributed to the modern English convention of marking short vowels by doubling a following consonant letter.
However, this kind of lengthening was not so regular for the vowels i or u. It seems that the verb put did not actually originally have /tː/ (the OED says the corresponding OE forms are probably something like *pūtian, *putian, pȳtan, potian). Similarly, the verb nut is derived from the noun, which corresponds to an Old English form with singleton /t/, like hnutu. Despite the etymologies of these words, putting and nutting are spelled with -tt- in present-day English.
The verb vet is ultimately derived from Latin veterīnārius, with singleton /t/, but we write vetting and vetted.
If "emitting" does phonologically contain double tt, then "emit" probably does also
Some linguists have considered the possibility that the English sound system might contain some very abstract elements and processes: for example,
a process that shortens underlying aː or lengthens underlying a to produce the surface alternation between æ and eɪ in certain words like sane/sanity;
a word-final vowel that has zero as its surface realization (supposedly present in words like ellipse);
a process that turns underlying ng into ŋ in certain contexts (e.g. in word-final position)
In theories like this, I have seen reference to the concept of "virtual geminate" consonants that are supposed to explain the otherwise exceptional pronunciations of certain words; I think that for example, the final stress of words like omit, emit, permit could be interpreted as a sign that they "underlyingly" end in tt even in the present-day sound system of English. But I don't think many people actually believe that theories like this are true. Even if it were true, this tt would be part of the root: words ending in tt would take the same inflectional suffixes as words ending in single t.