The free online dictionary Wikipedia is called ウィキペディア in kana.
Why is it not called ワィキペディア? Which string of kana provides a pronunciation closer to the English word?
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ウィ is the standard way of transcribing [wi] or [wɪ]. Similarly ウェ is used for [wɛ] (for example website → ウェブサイト) and ウォ for [wɒ] or [wɔ] (for example wombat → ウォンバット or walkman ウォークマン).
Here ウ is used to emulate the [w] sound and ィ is a small kana, indicating the vowel. The small ィ also makes ウィ into a digraph (same length as single full-sized kana).
This is similar to ファ フィ フェ フォ which are used to transcribe/transliterate words containing F, where フ is used to emulate the [f] sound (which actually becomes [ɸ]) and the small kana ァィェォ indicates the vowel, giving for example ファ [ɸa].
It doesn't make sense to use ワ here and it's actually not clear how ワィ should be pronounced (small kana aren't used with full-sized kana from the ア row, カサタナハマヤラワ).
See for example the Extended katakana section in https://en.wikipedia.org/wiki/Transcription_into_Japanese.
A closer transcription of the English //ˌwɪkɪˈpiːdɪə// would be ウィキピーディア, but transcription into Japanese is often a compromise between spelling and pronunciation.
Rather than providing a pronunciation closer to the English word, it provides a description of the Japanese pronunciation.
The Japanese W sound is essentially spoken by moving from a うto a second vowel. I. E. わ starts as if you are going to say う and then your tongue moves into position to say あ. This is different from English W which uses more rounded lips and is hence also why Japanese people have trouble saying words like woman, wood, wool.
ウィ is actually a more accurate representation of how Japanese people would pronounce this word.
On the other hand, think about how romaji is used to represent Japanese words.
But these are more familiar sounds to English speakers and provide a way for them to speak within the realm of English phonotactics.
The same thing happens even between English and German.
Apparently, even historical kana spelling used ウ rather than ワ, as in ウヰスキー (still kept in company names such as ニッカウヰスキー. So where would the usage of ワ come from?
As you explicitly refer to Wikipedia, I found a couple of discussions on the kana displayed in the Wikipedia logo, such as this (from 2006):
The current Wikipedia logo originally contained the character ヰ, obviously a misinterpretation of the Japanese spelling of the word wikipedia, and was shortly after changed to what looks like クィ, which is a nonsensical character combination and was probably another effort at spelling wikipedia in Japanese with a ワィ, which is also nonsensical. (The correct character-combo should be ウィ.)
The other answers have pointed out that ウィ is the preferred kana combination, which also happens to be the standard suggestion if I type
wi on my Japanese keyboard in kana mode. This answer will attempt to state why ワィ would be a bad choice.
The most common kana to be followed by small kana are those of the /i/ series used together with ゃゅょ to form palatalised consonants. Historically, the next common would be く or ぐ followed by a /wa/-series small kana (わゑゐ; I am not sure if を was used for this purpose, too) to denote a labialised consonant which appeared and mostly disappeared in Middle Japanese – Kwansei Gakuin University being one of the last remnants of this and still sometimes spelt くゎんせい in hiragana depending on the intended pronunciation.
The choice of /i/ and /u/-series is not arbitrary because the vowel sound these series have is closest to the semivowel that is intended: /i/ is the closest vowel to the semivowel /j/ and likewise /u/ is the closest vowel to the semivowel /w/.
In modern times, further varieties of small kana have been added to accomodate the pronunciation of (often English) loanwords which do not fit into common Japanese phonology. Here typically, the first kana provides a consonant sound while its vowel is to be disregarded while the small kana provides the actual vowel. Sometimes, as in ファ/フィ/フェ/フォ, there is only one consonant kana to choose from. In other times, there are many such kana yet only the one with the closest vowel is used: thus, we see ティ and トゥ for /ti/ and /tu/ but not テゥ or トィ or any variant with タ. Using the closest vowel provides the additional benefit that if you happen to misrender or misread the small vowel as a standard-sized one, the resulting pronunciation still approximates the intended one.
For /wi/, the kana sequence ワィ might suggest itself based on the ‘there is only one consonant choice’ rule – but then note that English w like its Japanese counterpart isn’t actually a full consonant but a semivowel. This is reflected in there being no traditional kana for /wu/ (or /yi/) although some were assigned in the Meiji period and also by the use of /wa/ series kana to signify labialisation where present as outlined above. Therefore, instead of invoking a /wi/ style pronunciation it would be much more obvious to use this sequence of kana to represent a /wai/ style pronunciation. Indeed, the sequence ワイ (using a full-size イ) is present in a number of loanwords such as ワイン (wine) or ワイヤレス (wireless) where the English letter combination wi is pronounced as a /wai/ diphthong.