From the "Earliest Known Uses of Some of the Words of Mathematics" pages ...
According to Cajori (1919, page 175), “The words ‘abscissa’ and ‘ordinate’ were not used by Descartes. ... The technical use of ‘abscissa’ is observed in the eighteenth century by C. Wolf and others. In the more general sense of a ‘distance’ it was used earlier by B. Cavalieri in his Indivisibles, by Stefano degli Angeli (1623-1697), a professor of mathematics in Rome, and by others.”
Abscissa is found in Latin in 1656 in Exercitationum Mathematicarum by Frans van Schooten. See page 285. [Bill Stockich]
According to Cajori (1906, page 185), “The term abscissa occurs for the first time in a Latin work of 1659, written by Stefano degli Angeli (1623-1697), a professor of mathematics in Rome.” A footnote attributes this information to Moritz Cantor.
Abscissa was used in Latin by Gottfried Wilhelm Leibniz (1646-1716) in “De linea ex lineis numero infinitis ordinatim ductis inter se concurrentibus formata....,” Acta Eruditorum 11 1692, 168-171 (Leibniz, Mathematische Schriften, Abth. 2, Band I (1858), 266-269).
According to Struik (page 272), “This term, which was not new in Leibniz's day, was made by him into a standard term, as were so many other technical terms.”
For the word in English the OED cites a passage from 1706: H. Ditton An Institution of Fluxions p. 31 “’Tis required to find the relation of the Fluxion of the Ordinate to the Fluxion of the Abscisse.”
Cajori (1906, page 185) writes: "The Latin term for 'ordinate,' used by Descartes comes from the expression lineae ordinatae, employed by Roman surveyors for parallel lines.
Cajori (1919, page 175) writes: "In the strictly technical sense of analytics as one of the coördinates of a point, the word "ordinate" was used by Leibniz in 1694, but in a less restricted sense such expressions as "ordinatim applicatae" occur much earlier in F. Commandinus and others."
Leibniz used the phrase "per differentias ordinatarum" in a letter to Newton on June 21, 1677 (Scott, page 155).
Leibniz used the term ordinata in 1692 in Acta Eruditorum 11 (Struik, page 272).
For the word in English the OED has a passage from 1706: H. Ditton An Institution of Fluxions p. 31 “'Tis required to find the relation of the Fluxion of the Ordinate to the Fluxion of the Abscisse.”
"Earliest Known Uses..." doesn't have an entry for "applicate". The Math Forum has a Dr. Math thread about the term, referencing PlanetMath as a source. I can't seem to find the PM entry, but here's part of it as quoted in the thread:
The [...] name “applicate” is rare in English, but its equivalents in continental European languages, as “die Applikate” in German and “aplikaat” in Estonian, are more known.
There's no mention of how that particular term originated, however. (As with Dr. Math respondent, I'd never heard the term.)