The Greek phrase is μὴ ὀνειδίζοντος, which is the negative particle μὴ followed by the present tense, active voice, participle ὀνειδίζοντος declined in the singular number, genitive case. ὀνειδίζοντος is conjugated from the verb ὀνειδίζω. This verb occurs 10 times in the Textus Receptus (Estienne, 1551). BDAG defines the verb ὀνειδίζω as,
① to find fault in a way that demeans the other, reproach, revile, mock, heap insults upon
② to find justifiable fault with someone, reproach, reprimand,
In addition, BDAG notes,
A special kind of reproach is the suggestion of reluctance that too often accompanies the giving of a gift (Sextus 339 ὁ διδοὺς μετʼ ὀνείδους ὑβρίζει; difft. Plut., Mor. 64a; s. also Sir 20:15; 41:25.—ὀν. can also mean charge or reproach someone with someth., a kind of verbal extortion, with the purpose of obtaining someth. from a pers., e.g., Maximus Tyr. 5, 7h τῷ θεῷ the building of a temple); God does not do this Js 1:5.
The Greek text states,
εἰ δέ τις ὑμῶν λείπεται σοφίας αἰτείτω παρὰ τοῦ διδόντος θεοῦ πᾶσιν ἁπλῶς καὶ μὴ ὀνειδίζοντος καὶ δοθήσεται αὐτῷ
James writes that God gives to all men [ἁπλῶς]. The adverb ἁπλῶς means "simply." It is related to the adjective ἁπλοῦς, meaning "single, simple." In this context, ἁπλῶς suggests that God gives simply, without strings attached.
Henry Alford notes,
but we must interpret by what follows, and understand it of simply giving, and adding nothing afterwards which may take off from the graciousness of the gift)...
Imagine if you ask someone for some kind of assistance, whether labor or financial. The person obliges, but on condition. God does not give wisdom in this manner. Rather, He gives wisdom simply, without condition, to all who ask.
As for μὴ ὀνειδίζοντος, imagine you ask someone for something, and they respond, "Again? I just gave you some." The individual is not only reluctant to give you what you requested, but you are also shamed/ insulted in the process. You can also imagine the mannerisms and tone of voice that would accompany such a response.
For other verses related to the general context, see:
Arndt, W., Danker, F. W., & Bauer, W. (2000). A Greek-English lexicon of the New Testament and other early Christian literature (3rd ed.) (710). Chicago: University of Chicago Press.