My vowel phonemic inventory would be just /a/, /e/, /o/ and /u/ with no /i/ sound. Is this possible or not?
This would be very unstable in a human language. For an example of what happens when this vowel is missing, let's look at English's Great Vowel Shift. One of the first things to change was /i:/ becoming /aj/, leaving it without /i/. Within less than a hundred years, /e:/ had migrated to take its place.
In your specific case, I would expect /e/ -> /i/, /a/ -> /e/, and /o/ -> /ɒ/ in fairly short order.
This is because vowels like to spread out on the vowel chart and become more distinct to improve understanding. Because of the limitations of your mouth making it difficult to have highly fronted low vowels, the general rule is that you will have more high vowels than mid or low vowels. Your vowel system violates this universal (with more mid vowels than high vowels).
For a constructed language, this is definitely possible. It is not a natural choice but not completely unseen in natural languages, according to PHOIBLE 92% of the sampled languages contain the vowel /i/.
A vowel system of /a e o u/ would certainly be unusual and unstable, and appears to be unattested, but lack of /i/ is certainly attested. Marshallese, for instance, has the thoroughly strange vowel inventory of /a ɜ ɘ ɨ/ (though [i] is present phonetically, as an allophone of /ɨ/). Kalam has a vowel system of /a e o/ (though [i] is again present phonetically, as an allophone of /j/). And, depending on which website you trust, Hixkaryana has either /a e ɨ o u/ or /æ e ɯ ɔ u/.
you could, with conlangs you really can do anything. it doesn't need to make sense.
however you should probably ask yourself what your goal is with that. if you're going for naturalism, /i/ is one of the most universal sounds crosslinguistically, a language without /i/ is practically unheard of... even languages that don't have an /i/ in the inventory will usually have it as an allophone of something. of course, nothing is stopping you from making a naturalistic lang with no /i/ still... it would be a weird trait, but overall, hardly any real life language is 100% naturalistic, and they tend to have strange quirks. even if this one is one of a kind, if you are going for naturalism in other areas, it could probably be passable.
if you want to make an auxlang, then again, lack of /i/ would be strange given it is easy to pronounce, and seen in practically all languages.
if you want an artlang and you simply dont like the sound of /i/ then there's not much otherwise stopping you. the same if your language is for a non-human phonology.