Why is ggpredict() not outputting robust standard errors?

Strange, I cannot reproduce your example. Do you have package sandwich installed? When I run your code, both CI and SE differ:

library(ggeffects)
data(mtcars)
reg <- glm(vs ~ mpg + cyl + disp + hp, mtcars, family="binomial")

ggpredict(reg, "mpg")
#> Data were 'prettified'. Consider using `terms="mpg [all]"` to get smooth
#>   plots.
#> # Predicted probabilities of vs
#> 
#> mpg | Predicted |     95% CI
#> ----------------------------
#>  10 |      0.70 | 0.02, 1.00
#>  12 |      0.61 | 0.03, 0.99
#>  16 |      0.39 | 0.05, 0.89
#>  20 |      0.21 | 0.03, 0.70
#>  22 |      0.15 | 0.01, 0.70
#>  24 |      0.10 | 0.00, 0.74
#>  28 |      0.05 | 0.00, 0.86
#>  34 |      0.01 | 0.00, 0.96
#> 
#> Adjusted for:
#> *  cyl =   6.19
#> * disp = 230.72
#> *   hp = 146.69
#> 
#> Not all rows are shown in the output. Use `print(..., n = Inf)` to show
#>   all rows.
ggpredict(reg, "mpg", vcov_fun = "vcovHC", vcov_type = "HC1")
#> Data were 'prettified'. Consider using `terms="mpg [all]"` to get smooth
#>   plots.
#> # Predicted probabilities of vs
#> 
#> mpg | Predicted |     95% CI
#> ----------------------------
#>  10 |      0.70 | 0.04, 0.99
#>  12 |      0.61 | 0.05, 0.98
#>  16 |      0.39 | 0.07, 0.84
#>  20 |      0.21 | 0.08, 0.48
#>  22 |      0.15 | 0.05, 0.38
#>  24 |      0.10 | 0.02, 0.37
#>  28 |      0.05 | 0.00, 0.45
#>  34 |      0.01 | 0.00, 0.63
#> 
#> Adjusted for:
#> *  cyl =   6.19
#> * disp = 230.72
#> *   hp = 146.69
#> 
#> Not all rows are shown in the output. Use `print(..., n = Inf)` to show
#>   all rows.

ggpredict(reg, "mpg") |> as.data.frame()
#> Data were 'prettified'. Consider using `terms="mpg [all]"` to get smooth
#>   plots.
#>     x  predicted std.error     conf.low conf.high group
#> 1  10 0.70259351  2.494497 1.747344e-02 0.9968235     1
#> 2  12 0.60535681  2.056297 2.653409e-02 0.9885484     1
#> 3  14 0.49899836  1.652617 3.757648e-02 0.9621327     1
#> 4  16 0.39273047  1.315624 4.677843e-02 0.8949872     1
#> 5  18 0.29573482  1.107940 4.568542e-02 0.7864790     1
#> 6  20 0.21424339  1.105050 3.031314e-02 0.7039801     1
#> 7  22 0.15041153  1.308311 1.344489e-02 0.6969606     1
#> 8  24 0.10310235  1.642913 4.571786e-03 0.7420843     1
#> 9  26 0.06945689  2.045381 1.353191e-03 0.8043664     1
#> 10 28 0.04622510  2.482931 3.730739e-04 0.8628956     1
#> 11 30 0.03050906  2.939941 9.893732e-05 0.9091607     1
#> 12 32 0.02002413  3.408591 2.564065e-05 0.9421399     1
#> 13 34 0.01309385  3.884672 6.548620e-06 0.9641321     1
ggpredict(reg, "mpg", vcov_fun = "vcovHC", vcov_type = "HC1") |> as.data.frame()
#> Data were 'prettified'. Consider using `terms="mpg [all]"` to get smooth
#>   plots.
#>     x  predicted std.error     conf.low conf.high group
#> 1  10 0.70259351 2.0829964 0.0383122772 0.9929123     1
#> 2  12 0.60535681 1.7346301 0.0487080617 0.9787027     1
#> 3  14 0.49899836 1.3958193 0.0606674610 0.9388743     1
#> 4  16 0.39273047 1.0756319 0.0728272983 0.8418890     1
#> 5  18 0.29573482 0.7968433 0.0809524471 0.6668767     1
#> 6  20 0.21424339 0.6182533 0.0750705886 0.4780690     1
#> 7  22 0.15041153 0.6315201 0.0488399517 0.3790411     1
#> 8  24 0.10310235 0.8274665 0.0222040963 0.3678585     1
#> 9  26 0.06945689 1.1135020 0.0083468764 0.3982797     1
#> 10 28 0.04622510 1.4367934 0.0028918349 0.4474840     1
#> 11 30 0.03050906 1.7771230 0.0009655520 0.5060871     1
#> 12 32 0.02002413 2.1263255 0.0003164239 0.5687938     1
#> 13 34 0.01309385 2.4806565 0.0001026144 0.6317112     1

^{Created on 2024-02-22 with reprex v2.1.0}