1. [. . .] I have requested they amend the quotation and submit it to us without further delay.
1. [. . .] I have requested that they amend the quotation and submit it to us without further delay.
Would please tell me which sentence is grammatically correct? Is the word "that" required in the sentence above?
Both your examples are correct without "Following the meeting," which, at least in my opinion, renders the present perfect ("have requested") unacceptable. "Following the meeting" is equivalent to "After the meeting." The present perfect doesn't work with such adverbials. But the past simple ("requested") would be fine:
1. Following the meeting, I requested they amend the quotation and submit it to us without further delay.
2. Following the meeting, I requested that they amend the quotation and submit it to us without further delay.
"That" can be omitted when it introduces a clause forming a complement to a verb. For example, "He said that it was fine" can be changed to "He said it was fine." Retaining "that" is advisable in such cases only when the subject of the complement clause could momentarily be mistaken for the direct object of the verb. In your example, that is impossible, since "they" is a subjective-case pronoun. But consider the following example pair:
3. He requested the forms be mailed immediately.
4. He requested that the forms be mailed immediately.
In (3), the reader may, for at least a fraction of a second, mistakenly parse "the forms" as the direct object of "requested," since it is possible to say "He requested the forms." If "that" is used, however, it is clear that "the forms" is to be parsed as the subject of a clause that is being introduced. Notice that in your example, the parsing "
I have requested they" would be ungrammatical. Because "they"
cannot be the direct object, "that" is not needed.