Did Thomas Aquinas's correspondent James of Tonengo become bishop of Vercelli?

score:11

Sort of, but he was not recognized by all the electors, some of whom elected another candidate. I can find no evidence that he was ever confirmed as bishop by the Holy See. In short, it's complicated...

Canon James of Tonengo, previously chaplain to Pope Urban IV (d.1264), was one of two Bishops of Vercelli elected by different factions of electors:

James of Tonengo was elected bishop of Vercelli, but only by a faction of the chapter. The other group supported and elected another candidate. The problem could not be promptly resolved by the Holy See because of a long period of vacancy (from November 1268 to September 1271).

Source: Pasquale Porro, 'Thomas Aquinas: a Historical and Philosophical Profile' (2012, translated 2016).

There does not appear to have been a sole, uncontested, Bishop of Vercelli for the period 1268 to 1273. This was partly due to there being no pope from November 1268 to September 1271, with a further complication being the deadlock among the electors which you mentioned in your question.

Although the Arcidiocesi Vercelli site, as cited by LangLangC in his answer, gives Aimone di Challant as bishop for the period 1268 to 1305, the period prior to 21 Dec 1273 seems unlikely if there were already two bishops elected by different factions. Also, French Wikipedia notes that Aimone (or Aime) was Bishop of Aosta from 30 August 1272 until 21st Dec 1273, and that he left Aosta on the latter date to take up the appointment at Vercelli. One of Wikipedia's sources for this is the Catholic Hierarchy site page on Bishop Aime de Chantal, though there is uncertainty over the appointment date of 1272. The same site's page for Bishop Events for 1273 also lists Aimone's appointment as 21 December 1273.

As to what the electors did with Thomas Aquinas advice, one can only assume that they did not go ahead and draw lots as the contested election does not appear to have been resolved until the 1273 appointment of Aimone di Challant. Thus, the electors appear to have followed his advice:

Thomas’s answer is...that it is not permissible to draw lots for the election of someone as bishop, or some other ecclesiastical office, because in this case the divine inspiration manifests itself through the agreement of the electors. Drawing lots in these circumstances would constitute a veritable sin against the Holy Spirit.

Source: Porro

However, there is some dispute over whether the document on this was connected to the contested Vercelli election due to uncertainty over dating of documents.

Other source:

Jean-Pierre Torrell, 'Saint Thomas Aquinas: the person and his work' (1993)