tl;dr literally no because it's an english term emerging from french and spanish roots; however similar words borrowed from old french were used at the time.

from the oed

modern French croisade (= Old French croisee ), Provençal crozada , Spanish cruzada , Italian crociata , medieval Latin cruciata (cruzata ), being in the various languages the feminine noun of action formed on past participle of cruciāre , crociare , cruzar , croiser to cross v., lit. [literally] a being crossed, a crossing or marking with the cross, a taking the cross

out 1600, the Spanish cruzada made its appearance under the forms crusada and crusado (see -ado suffix); a blending of this with croisade produced two hybrid forms, viz. croisado (-ada ), with French stem and Spanish ending, frequent from c1611 to 1725, and crusade , with Spanish stem and French ending, mentioned by Johnson, 1755,

i've seen multiple times "taking up the cross" in some speeches from the time. So the term crusade comes into english too late to have been used by the Crusaders but antecedents of the phrase were used. Having not really read the primary sources outside of translation i can't really tell what exact words were used but it's clear the roots of the word crusade were used at the time to describe the conflicts. Some similar middle english terms (take from french) were used such as croiserie

e.g. Chronicle of Robert of Glouster (13th century)

Þe pope sende croiserie in to þe holi lond.