James was an apostle (Galatians 1:19) and one of the central figures of early Christianity (Galatians 2:9), as reflected by his prominence in Acts. Jesus' appearance to James after the resurrection (1 Corinthians 15:7) seems to be the best explanation for his conversion from his prior skepticism (Matthew 13:57, Mark 3:21-35, 6:4, Luke 8:19-21, John 7:5). So, why would none of the gospels mention Jesus' appearance to James, even though James was so prominent in early church history and Jesus' appearance to him had so much evidential value?
The appearance to Paul isn't mentioned either. But there's an easy explanation for its absence. The appearance to Paul wouldn't be mentioned because of its lateness. The gospel authors were covering an earlier timeframe. Accordingly, Luke does mention the appearance to Paul when he's covering a later period of time in Acts. It's significant, though, that nobody places the appearance to Paul earlier. Given that the gospels cover weeks worth of time after the resurrection (John 20:26, Acts 1:3), one or more of the gospel authors could easily have moved Paul's conversion into that timeframe if they had little or no interest in historical accuracy. Instead, they all exclude Paul. Paul's exclusion is significant, because of what I just mentioned, but it doesn't shed much light on the exclusion of James. Lateness, which is such a good explanation for Paul's exclusion, doesn't seem to be an adequate explanation for James' exclusion. The layout of 1 Corinthians 15:5-8 and the singling out of Paul in verse 8 suggest that the appearance to James was more early than late. Jesus still appeared to "all the apostles" after appearing to James (verse 7). James was a believer well before Paul in Acts (Acts 1:14), a fact also suggested by Paul's interactions with James as an established church leader in Galatians 1-2. It doesn't seem that the appearance to James had the sort of late quality that Paul's had. So, why isn't James included in the gospel narratives?
One possibility we can dismiss is the notion that the gospel authors didn't think there was an appearance to James. As Acts, 1 Corinthians, Galatians, and the letter of James illustrate, James was a prominent leader in the early church. There was widespread knowledge that he'd received a resurrection appearance (1 Corinthians 15:7). He was considered an apostle (Galatians 1:19, 2:9), which implies that he had seen the risen Jesus (1 Corinthians 9:1). The idea that all four gospel authors didn't believe in an appearance to James is absurd. Given that they surely did believe in an appearance to James, why didn't they mention it?
Maybe the appearance was of too ordinary a nature. It didn't involve anything as narratively appealing as, say, the encounter on the road to Emmaus or the appearance to Thomas. But other appearances mentioned in the gospels are relatively ordinary. Ordinariness may be a partial explanation for the exclusion of James, but I doubt that it's the only explanation.
The best explanation I'm aware of is the prior prominence of Jesus' earliest followers. Because Jesus' earliest followers (in contrast to later ones, like James) had done the right thing by honoring Jesus even prior to his resurrection, perhaps it was considered fitting to focus on them in the resurrection accounts. And since the gospels had given so much attention to Jesus' earliest followers in the pre-resurrection narratives, there would be some merit in limiting the resurrection accounts to them. That would give the narratives more continuity with what came before.
There are some important implications to James' exclusion. For one thing, it demonstrates a significant level of restraint on the part of the gospel authors. They weren't trying to be exhaustive, nor were they necessarily even attempting to include the best evidence they had for Jesus' resurrection. When skeptics suggest that something like the appearance to the more than 500 (1 Corinthians 15:6) must not have actually happened, since none of the gospel authors mention it, they're bringing some false assumptions to the text. And the absence of James casts doubt on skeptical speculations about how the resurrection narratives supposedly were invented to contribute to various power struggles in the early church. As N.T. Wright mentions:
"In particular, if it is true that stories of people meeting Jesus were invented in order to legitimate leaders in the early church, it is remarkable that we hear nothing, throughout the gospel stories, of James the brother of Jesus….Why does he, too, not run a race against Peter [as in John 20:3-8]? Would that not have been a convenient fiction to clothe early ecclesial power struggles?" (The Resurrection Of The Son Of God [Minneapolis, Minnesota: Fortress Press, 2003], 610)
The exclusion of James (and Paul and whoever else) should also remind us of the significance of the earliest followers of Jesus. Witnesses like the women at the tomb, Peter, and Thomas are often dismissed as too Christian. They believed in Jesus before he died. Their experiences and their testimony pertaining to the resurrection aren't worth much. But somebody like Mary Magdelene or Peter doesn't have to have the same significance as a James or a Paul in order to have a lot of significance nevertheless.