What does つまづい mean?

Question

What's the meaning of つまづい in this context?

たとえ何があっても
つまづいても迷っても

Answer 1

You're parsing it incorrectly. The second line should be

つまづいて・も・迷って・も

So つまづいて is the 〜て form of つまづく, which @rdb pointed out is probably an older spelling of つまずく【躓く】, which means "to trip/stumble".

Answer 2

Let me add a little about the spelling つまづく. It is an alternate (secondary) spelling of つまずく, and not necessarily an old spelling as stated in other answers.

This verb was etymologically a compound word made of つめ and つく with a vowel mutation (つめ→つま) and rendaku (つく→づく). In the historical kana orthography, it was written as つまづく, reflecting the fact that the latter part came from つく.

In the modern kana orthography, most uses of letter づ were replaced with ず to match the orthography with pronunciation. One of the exceptions is that づ arising from rendaku has been retained. If you think of the origin, づ in つまづく indeed arises from rendaku and therefore the word would have been written as つまづく even in the modern orthography. However, because it is not obvious from its meaning that this verb is a compound word, it was decided in the official recommendation that the primary spelling of this verb should be つまずく, with a remark that it can be alternately written as つまづく. The recommendation also lists other examples of words of this kind.

So, つまづく is not only the spelling in the historical orthography but also a permitted spelling in the modern orthography, although the primary spelling in the modern orthography is つまずく.

Answer 3

http://dictionary.goo.ne.jp/leaf/je2/50364/m1u/%E8%BA%93%E3%81%8F/

It's just old 仮名遣い.