Koine Greek has second-person and third-person imperative forms, each of which can be expressed by a single verb. In the case of ἔρχομαι:
However, English truly only has a second-person imperative. For example, “Come!” For the first and third-person, known as the cohortative (adhortative) and jussive, respectively, English uses auxiliary verbs in conjunction with the primary verb.
In Modality in English: Theory and Description, it states,1
...in addition, the prototypical English imperative (the second person imperative) contains the base form of the verb, as in Be quiet! In some approaches, such second person imperatives are called orders or commands, and in narrow definitions of imperatives, the three terms are synonymous. However, under a wider (and I believe more useful) definition, imperatives are not confined to second person imperatives, for there exist also first person imperatives (adhortatives) and third person imperatives (or jussives). In English, two of these — first and third person imperatives — can occur with or without let. In other words, a common way to express imperatives in English is by means of a periphrastic construction involving let.
In John 7:37, the King James Version employed “let” as an auxiliary verb in conjunction with the primary verb “come” to translate the Greek third-person imperative.
On the word “let,” the Oxford English Dictionary states,2
The imperative with noun or pronoun as object often serves as an auxiliary, forming the equivalent of a first or third person of the verb which follows in the infinitive. Also (U.S. colloquial) in irregular phrase let's you and me (or you and I, or us): let us (do something).
One should not confuse that use of “let” as an auxiliary to express an imperative with another use in the sense of “not to prevent; to suffer, permit allow.”3 For example, such a use occurs in the following scripture,
15 And when the ship was caught, and could not bear up into the wind, we let her drive.
John 7:37 is not saying, “If anyone thirst, he may come to me and drink.” Or, “...permit him to come to me.” Rather, it is ordering or commanding the thirsty person, “If anyone thirst, come to me and drink!” It is not a request; it is a command (imperative).
Furthermore, the verb ἔρχομαι is conjugated in the middle-voice because it is a deponent (verb, a verb which lacks a form in the active voice),4 not because of some implied notion of permission or volition.
In summary, the Lord Jesus Christ is commanding whoever is thirsty to come to him. “Let” is only being used periphrastically to express the third-person imperative which English has no usual means to express. It is not being used to imply allowance or volition, although it is obvious that one must come voluntarily. Can someone come to something without choosing to move their own self? No. Otherwise, they are not coming but being brought.
1 p. 316
2 OED online, “let, v.1,” 14., a.
3 OED online, “let, v.1,” II., 12., a.
4 Mounce, § 18.15
Bourdin, Philippe. “On the “great modal shift” sustained by come to VP.” Modality in English: Theory and Description. Ed. Busuttil, Pierre; Salkie, Raphael; van der Aiwera, Johann. Berlin: De Gruyter, 2009.
Mounce, William D. Basics of Biblical Greek. 4th ed. Grand Rapids: Zondervan, 2019.