[Edited to add: Main solution already found, and appended below.]
Especially when trying to enter answers for certain vocab words (usually loan words), sometimes we need to type uncommon but legitimate kana constructions like てぃ or, more accurately, katakana ティ (although usually when appropriate, the text entry will automatically convert hiragana to katakana on Enter). For example, パーティー, ボランティア, and actually quite a few others I haven’t encountered yet (Bunpro Vocab search for ティ).
There are easy-to-type automatic conversions for more-common constructions, like ‘ju’ auto-converting to ‘じゅ’, and there are even some auto-conversions for uncommon constructions (mostly for loan words), like ‘fi’ → ‘ふぃ’, and ‘vi’ → ‘ヴぃ’.
Meanwhile there are also some shortcuts for very-common single-kana like ‘si’ short-cutting to ‘し’, instead of having to type three characters ‘shi’. Likewise, there’s ‘tu’ for ‘tsu’ → ‘つ’. And, most relevant for this post, there’s ‘ti’ instead of ‘chi’ → ‘ち’.
So, if you want to type てぃ, well ‘ti’ is unfortunately already taken! ‘ti’ converts to ‘ち’, rather than the much-less-common ‘てぃ’.
So, I’m hoping that there is an easy, short way that already exists to type てぃ using standard romaji-input IME (or the browser-script equivalent used by Bunpro and other Japanese learning websites).
I do know that there is an ‘extended’ way to type てぃ, which is to enter each sub-kana separately, with ‘te’ for て and ‘xi’ or ‘li’ for ‘small’ or ‘little’ ぃ, thus taking four characters ‘texi’ or ‘teli’ to get てぃ. But is there an existing shortcut that only takes two (or possibly just three?) characters which results in a proper ‘てぃ’?
Side question: Is てぃ the only (relatively) common combination for loanwords that (seemingly) requires four characters / extended input to type it, especially for the Vocab currently in use on Bunpro? Or are there others? I can’t think of any off the top of my head, but maybe I just haven’t come across them yet. (Also, are ‘x_’ and ‘l_’ the only ways to output these kinds of ‘small’/‘little’ kana, or is there any other way/keyboard-combo?)
If other folks can mention some other clunky combos used in Bunpro Vocab, I’ll edit this post to include them. Thanks!
Solutions so far!
Thanks to @Slysoft , @Pablunpro , and @Humin for quickly coming to the rescue with these answers, which happen to use three characters instead of only two, but at least not four! 
(in あいうえお order):
For てぃ and similar
てぃ
- てぃ itself can be entered with ‘thi’.
て (general case) ・ て_ = th_ (= tel_ = tex_)
| 平仮名 | 片仮名 | romaji | Vocab? |
|---|---|---|---|
| てゃ | テャ | tha | |
| てぃ | ティ | thi |  |
| てゅ | テュ | thu | |
| てぇ | テェ | the | |
| てょ | テョ | tho |
で (with 濁点) ・ で_ = dh_ (= del_ = dex_)
| 平仮名 | 片仮名 | romaji | Vocab? |
|---|---|---|---|
| でゃ | デャ | dha | |
| でぃ | ディ | dhi | |
| でゅ | デュ | dhu | |
| でぇ | デェ | dhe | |
| でょ | デョ | dho |
Additional
There’s probably some official table somewhere compiling all of these. But anyway, here’s more that @Humin found (although probably none of these appear in Bunpro Vocab, according to @mrnoone ).