These Google Ngrams appear to show that
(1) both variants are in use
(2) the “there are a total of" version is about twice as common
(3) popularity has reversed since about 1970.
People answering 'it should be ...' are choosing one of the conflicting 'rules' rather than another (and results show that they're hardly worthy of the name 'rule').
“There is a total of 378 vehicles” uses what is known as 'grammatical concord' (there being a single total).
“There are a total of 378 vehicles” uses (at least equally correct) notional concord; compare 'If a majority of the members vote for the bill ...'. I'd consider 'There is a total of 378 vehicles waiting for replacement timer chains' extremely awkward to unacceptable. 'There are a total of ...' is idiomatic for 'There are, in total, ...' when used with further expressions such as 'waiting for replacement timer chains' or 'which need new tyres'.