The answer to this question will depend on the level of transcription used. In very broad transcription of English, the answer yes. For more narrow transcription, however (including vowel length and offglides), the short answer is no. In general, there is no English word which can end in a short unstressed monophthong that is not [ə], or a schwa-like sound.
This is due to several phonological processes in English:
(1) A vowel lengthening process that applies to tense vowels in an open syllable. In a simplified case this would include all words terminating in a tense vowel.
(2) A constraint that forbids open syllables in English terminating in a lax vowel. These are also known as checked vowels.
(3) A vowel reduction process that applies to unstressed vowels. These vowels become neutralized (information is lost), are no longer contrastive, and in general reduced to a schwa-like sound, which is canonically transcribed as [ə] in English.
To see why this plays out, let us consider the possible cases of a word ending in a short unstressed monophthong and why these phonological processes make it impossible.
(a) Suppose we have some word which ends in a tense vowel V. Tense vowels are lengthened when in open syllables according to (1). Thus, no matter how stress is assigned, V will be realized as V: (or VV, whichever you prefer).
(b) Suppose we have some word which ends in a lax vowel V. According to (2), lax vowels are checked in English, so there can be no open syllable ending in a lax vowel. Syllables which do violate English phonotactics (therefore they are not English!).
However, it does seem other unstressed vowels besides schwa can terminate a word. From a phonetics course:
three weak, unstressed vowels Schwa, KIT and FOOT
do occur at the end of words such as teacher, China, happy, to.
In particular, words like < happy > and < copy > are transcribed as ending in a short unstressed tense high front vowel [i]:
['hæpi], ['kɒpi] 
Nevertheless, tense vowels (in English) are actually often transcribed as having a glide appended, matching the "place" of the vowel.
< see > [sij] 
< who > [hu:w] 
For the word , this corpus of English by Heather Goad (Associate Professor of Linguistics at McGill University) also transcribes it with the final sound being the palatal glide [j]. Here are eight such instances from her corpus.
(the same also holds true for < you >, there over 200 instances of it being transcribed as [juw] which is not the same as [ju])
From these two sources, then, < happy > does not terminate in a short unstressed monophthong and should be transcribed as [hæpij]. The appended glides are called offglides, and sometimes are put in the superscript of the vowel with the offglide. Vowels with offglides are not monophthongs but diphthongs.
[i:] ~ [iy] ~ [ɪY] alternate in English, but [i] is absent altogether. This again lends evidence that < happy > does not end in an unstressed short monophthong but a short unstressed diphthong.
Note additionally that vowel length is not contrastive, which makes sense when we realize that vowel length is predictable in English. Contrastive features cannot be predicted, since they are used to encode lexical information.
As a disclaimer we must remember that the IPA symbols are abstract and idealized representations of sounds present in natural language. Thus there may be some speakers whose schwa might not accurately be transcribed as [ə] but perhaps closer to [ɨ] or even [ʌ]. Regardless of the specifics of the realization, none of these are contrastive, e.g. saying < a boot > [ʌbut] is the same as [əbut]. However, if we want to stress the definite article, the phrase gets realized as [ebut], e.g. It was a boot that John lost (and not a foot). This is because schwa in general cannot be assigned stress and from (3) unstressed vowels are generally reduced to schwa.
Applied Phonetics & Phonology: English Vowels. Tom Payne, TESOL at Hanyang University, 2007. http://pages.uoregon.edu/tpayne/APP2007/APPSession03-4-2007-Payne-print.pdf
ENS101G Phonetics I: Lax vowels. Pétur Knútsson, 2008. https://notendur.hi.is/peturk/KENNSLA/02/TOP/VowelsLax.html
ENS101G Phonetics I: Stress and weak vowels (Week 5, powerpoint). Pétur Knútsson, 2008. https://notendur.hi.is/peturk/KENNSLA/02/02schedule.html
Goad, Heather (2010). English-Goad: Online Corpus of Phonological Development. McGill University. ISBN 1-59642-438-9. Web access: http://childes.talkbank.org/data/
Stockwell, Robert and Donka Minkova. Explanations of Sound Change: Contradictions between Dialect Data and Theories of Chain Shifting. Web access: http://www.ling.ohio-state.edu/~ddurian/AWAC/Stockwell%20Minkova%201999.pdf