Converting logistic regression coefficients and standard errors into odds ratios is trivial in Stata: just add , or to the end of a logit command:

. use "http://www.ats.ucla.edu/stat/data/hsbdemo", clear

. logit honors i.female math read, or

Logistic regression                             Number of obs     =        200
LR chi2(3)        =      80.87
Prob > chi2       =     0.0000
Log likelihood = -75.209827                     Pseudo R2         =     0.3496

------------------------------------------------------------------------------
honors | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
female |
female  |   3.173393   1.377573     2.66   0.008      1.35524    7.430728
math |   1.140779   .0370323     4.06   0.000     1.070458     1.21572
read |   1.078145    .029733     2.73   0.006     1.021417    1.138025
_cons |   1.99e-06   3.68e-06    -7.09   0.000     5.29e-08    .0000749
------------------------------------------------------------------------------


Doing the same thing in R is a little trickier. Calculating odds ratios for coefficients is trivial, and exp(coef(model)) gives the same results as Stata:

# Load libraries
library(dplyr)  # Data frame manipulation
library(broom)  # Convert models to data frames

# Use treatment contrasts instead of polynomial contrasts for ordered factors
options(contrasts=rep("contr.treatment", 2))

# Load and clean data
df <- read_csv("http://www.ats.ucla.edu/stat/data/hsbdemo.csv") %>%
mutate(honors = factor(honors, levels=c("not enrolled", "enrolled")),
female = factor(female, levels=c("male", "female"), ordered=TRUE))

# Run model
model <- glm(honors ~ female + math + read, data=df, family=binomial(link="logit"))
summary(model)
#>
#> Call:
#> glm(formula = honors ~ female + math + read, family = binomial(link = "logit"),
#>     data = df)
#>
#> Deviance Residuals:
#>     Min       1Q   Median       3Q      Max
#> -2.0055  -0.6061  -0.2730   0.4844   2.3953
#>
#> Coefficients:
#>               Estimate Std. Error z value Pr(>|z|)
#> (Intercept)  -13.12749    1.85080  -7.093 1.31e-12 ***
#> femalefemale   1.15480    0.43409   2.660  0.00781 **
#> math           0.13171    0.03246   4.058 4.96e-05 ***
#> read           0.07524    0.02758   2.728  0.00636 **
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for binomial family taken to be 1)
#>
#>     Null deviance: 231.29  on 199  degrees of freedom
#> Residual deviance: 150.42  on 196  degrees of freedom
#> AIC: 158.42
#>
#> Number of Fisher Scoring iterations: 5

# Exponentiate coefficients
exp(coef(model))
#>  (Intercept) femalefemale         math         read
#> 1.989771e-06 3.173393e+00 1.140779e+00 1.078145e+00

# Exponentiate standard errors
# WRONG
ses <- sqrt(diag(vcov(model)))
exp(ses)
#>  (Intercept) femalefemale         math         read
#>     6.364894     1.543557     1.032994     1.027961


Calculating the odds-ratio adjusted standard errors is less trivial—exp(ses) does not work. This is because of the underlying math behind logistic regression (and all other models that use odds ratios, hazard ratios, etc.). Instead of exponentiating, the standard errors have to be calculated with calculus (Taylor series) or simulation (bootstrapping). Stata uses the Taylor series-based delta method, which is fairly easy to implement in R (see Example 2).

Essentially, you can calculate the odds ratio-adjusted standard error with $\sqrt{\text{gradient} \times \text{coefficient variance} \times \text{gradient}}$, and since the first derivative/gradient of $e^x$ is just $e^x$, in this case the adjusted standard error is simply $\sqrt{e^{\text{coefficient}} \times \text{coefficient variance} \times e^{\text{coefficient}}}$ or $\sqrt{(e^{\text{coefficient}})^2 \times \text{coefficient variance}}$

Doing this in R is easy, especially with broom::tidy():

model.df <- tidy(model)  # Convert model to dataframe for easy manipulation
model.df
#>           term     estimate  std.error statistic      p.value
#> 1  (Intercept) -13.12749111 1.85079765 -7.092883 1.313465e-12
#> 2 femalefemale   1.15480121 0.43408932  2.660285 7.807461e-03
#> 3         math   0.13171175 0.03246105  4.057532 4.959406e-05
#> 4         read   0.07524236 0.02757725  2.728422 6.363817e-03

model.df %>%
mutate(or = exp(estimate),  # Odds ratio/gradient
var.diag = diag(vcov(model)),  # Variance of each coefficient
or.se = sqrt(or^2 * var.diag))  # Odds-ratio adjusted
#>           term     estimate  std.error statistic      p.value           or
#> 1  (Intercept) -13.12749111 1.85079765 -7.092883 1.313465e-12 1.989771e-06
#> 2 femalefemale   1.15480121 0.43408932  2.660285 7.807461e-03 3.173393e+00
#> 3         math   0.13171175 0.03246105  4.057532 4.959406e-05 1.140779e+00
#> 4         read   0.07524236 0.02757725  2.728422 6.363817e-03 1.078145e+00
#>       var.diag        or.se
#> 1 3.4254519469 3.682663e-06
#> 2 0.1884335381 1.377536e+00
#> 3 0.0010537198 3.703090e-02
#> 4 0.0007605045 2.973228e-02


This can all be wrapped up into a simple function:

get.or.se <- function(model) {
broom::tidy(model) %>%
mutate(or = exp(estimate),
var.diag = diag(vcov(model)),
or.se = sqrt(or^2 * var.diag)) %>%
select(or.se) %>% unlist %>% unname
}

get.or.se(model)
#>  3.682663e-06 1.377536e+00 3.703090e-02 2.973228e-02


Same results in both programs! 