Yes, in the context of the quoted language from Lampleigh v. Braithwait, the word suit almost certainly means "A petition or entreaty made to a person in authority."
Long before suit meant "a formally submitted legal proceeding," it had various senses in English law, including, according to Black's Law Dictionary (1968) "The witnesses or followers of the plaintiff." The same source quotes one "Abbott" (perhaps Benjamin Vaughan Abbott) at some length in connection with the various meanings that suit had in "old English law":
Old books mention the word in many connections which are now disused,—at least in the United States. Thus, "suit" was used of following any one, or in the sense of pursuit; as in the phrase "making fresh suit." It was also used of a petition to the king or lord. "Suit of court" was the attendance which a tenant owed at the court of his lord. "Suit covenant" and "suit custom" seem to have signified a right to one's attendance, or one's obligation to attend, at the lord's court, founded upon a known covenant, or an immemorial usage or practice of ancestors. "Suit regal" was attendance at the sheriff's town or leet, (his court.) "Suit of the king's peace" was pursuing an offender,—one charge with breach of the peace, while "suithold" was a tenure in consideration of certain services to the superior lord.
So a suit was originally (in legal proceedings) a retinue or entourage accompanying a plaintiff to court (a term with much in common with suite); and as a verb it was closely tied to the notion of pursuit. But from there it came to mean a petition to one's king or lord, suggesting that the wording quoted by the OP—"a suit or request of the party"—is treating suit and request as synonyms.
Also of interest is this note in Black's Law Dictionary about the legal meaning of the word suitor:
SUITOR. A party to a suit or action in court. In its ancient sense, "suitor" meant one who was bound to attend the county court; also one who formed part of the secta ["the plaintiff's suit or following"].
This note suggests that suitor in the the sense of "wooer" may have arisen from suitor in the sense of one required to be in attendance at court; coincidentally (or not), court also became a synonym for woo as a verb and for wooing as a noun.
Finally, I note that, according to Merriam-Webster's Eleventh Collegiate Dictionary (2003), the verb sue originally had the (now obsolete) meaning "to make a petition to or for." Today, besides meaning to initiate legal proceedings against, sue can mean (somewhat archaically) to woo, as in "He sued for her hand," and (still currently) to plead, as in "Following its defeat on the battlefield, the country sued for peace."