The code for the function is quite simple:

fgsea::plotEnrichment
function (pathway, stats, gseaParam = 1, ticksSize = 0.2) 
{
    rnk <- rank(-stats)
    ord <- order(rnk)
    statsAdj <- stats[ord]
    statsAdj <- sign(statsAdj) * (abs(statsAdj)^gseaParam)
    statsAdj <- statsAdj/max(abs(statsAdj))
    pathway <- unname(as.vector(na.omit(match(pathway, names(statsAdj)))))
    pathway <- sort(pathway)
    gseaRes <- calcGseaStat(statsAdj, selectedStats = pathway, 
        returnAllExtremes = TRUE)
    bottoms <- gseaRes$bottoms
    tops <- gseaRes$tops
    n <- length(statsAdj)
    xs <- as.vector(rbind(pathway - 1, pathway))
    ys <- as.vector(rbind(bottoms, tops))
    toPlot <- data.frame(x = c(0, xs, n + 1), y = c(0, ys, 0))
    diff <- (max(tops) - min(bottoms))/8
    x = y = NULL
    g <- ggplot(toPlot, aes(x = x, y = y)) + geom_point(color = "green", 
        size = 0.1) + geom_hline(yintercept = max(tops), colour = "red", 
        linetype = "dashed") + geom_hline(yintercept = min(bottoms), 
        colour = "red", linetype = "dashed") + geom_hline(yintercept = 0, 
        colour = "black") + geom_line(color = "green") + theme_bw() + 
        geom_segment(data = data.frame(x = pathway), mapping = aes(x = x, 
            y = -diff/2, xend = x, yend = diff/2), size = ticksSize) + 
        theme(panel.border = element_blank(), panel.grid.minor = element_blank()) + 
        labs(x = "rank", y = "enrichment score")
    g
}

So, I would just modify this to retain gene name in the object toPlot, and then use ggrepel::geom_text_repel() to add these genes to the plot. ggrepel is necessary because there is no way that you will be able to fit all labels on that plot.

Kevin