Vietnamese uses a high number of diacritics. Are there any languages that are even higher, in terms of either the maximum or the mean of the number of diacritics added to letters?
If we are speaking of current official spelling systems, I believe that Vietnamese wins the prize. Potential competitors would most likely be a language with a rich vowel system than included independent tone, phonation and nasalization contrast, which points to Ju|'hoansi and !Xóõ, which would have a higher maximum and probably a higher text frequency – if these languages were transcribed in IPA. However, Ju|'hoansi has been undergoing a progressive reduction in diacritics to the point that the 1994 orthography is diacritic-free (tone isn't marked). !Xóõ orthography likewise trades in diacritics for VC digraphs.
Ozumacín Chinantec has ten vowels (a, ä, e, ë, i, ɨ, o, ø, u, ʉ), which may be nasalized (indicated by an underscore), and nine tones (indicated by ˈ, ˊ, ˉ, ꜗ, ꜘ, ꜙ, ꜚ, ˜, ˋ after the syllable). This results in words such as "läꜙjë̱ë̱ꜙ", "ø̱ø̱hꜗ", or "hä̱ä̱˜". It seems that this beats even Vietnamese.
There is a non-standard language in northern Italy called Emilian which is divided into many sub-dialects. Some of the variations, like Bolognese, have at least one diacritic mark in almost every word. Example:
Al prémm sît bulgnaiṡ dla Raid däl Raid, dedichè a tótt i bulgnîṡ, zitadén e ariûṡ ed tótt i lè, e anc ai linguéssta e ai furastîr ch’i in vôlen savair de pió, e pò naturalmänt ai bulgnîṡ ch’i s én dscurdè la längua di nunón, mo ch’i la vôlen turnèr a inparèr!