Instead of a standard error why not compute the standard deviation of the posterior distribution of the OR? You can solve for it numerically very easily using an MCMC sampler.
Here is some R and JAGS code to do so.
################################################################
### ###
### Contingency Table Analysis for Obestity Data ###
### ###
################################################################
# Required Pacakges
library("ggplot2")
library("runjags")
library("parallel") # sets parallelization for MCMC
# set up the model
mod = 'model {
################################################
### Greater than 80th BMI Percentile ###
################################################
# marginal likelihood functions
n11_G ~ dbin(pi_one_G, n1_plus_G)
n21_G ~ dbin(pi_two_G, n2_plus_G)
#priors
pi_one_G ~ dbeta(1,1)
pi_two_G ~ dbeta(1,1)
# transformations
rho_G <- pi_two_G/pi_one_G
theta_G <- pi_two_G*(1-pi_one_G)/(pi_one_G*(1-pi_two_G))
delta_G <- pi_two_G-pi_one_G
}'
# set up the data
Dat = list(n1_plus_G = 108,
n2_plus_G = 88,
n11_G = 68,
n21_G = 44)
# Monitor these variables
Vars = c("pi_one_G","pi_two_G","rho_G","theta_G","delta_G")
# set up MCMC parameters
inits1=list(.RNG.name= "base::Wichmann-Hill",
.RNG.seed= 12341)
inits2=list(.RNG.name= "base::Marsaglia-Multicarry",
.RNG.seed= 12342)
inits3=list(.RNG.name= "base::Super-Duper",
.RNG.seed= 12343)
inits4=list(.RNG.name= "base::Mersenne-Twister",
.RNG.seed= 12344)
chains = 4
burn = 5000
samp = 10000
adapt = 5000
thin = 1
# parallel chains
cl = makeCluster(4)
# MCMC estimation
HjagsOut = run.jags(model = mod, monitor = Vars, data=Dat, n.chains=chains, thin = thin,
burnin = burn, sample = samp, adapt=adapt, method="rjparallel",method.options=list(cl=cl),
inits=list(inits1,inits2,inits3,inits4))
#summarize results
summary(HjagsOut)
plot(HjagsOut, layout=c(4,2))
The odds ratio parameter ($\theta_\text{G}$) is simply a function of the samples from the binomial parameters. This of course assumes certain study design. In this case I was looking at the difference in children's BMI percentile group (80th and above or below 80th) from a control and experimental group, pre and post intervention treatment. Therefore, the row totals (number of children in the experimental group and control group respectively) were fixed.
The beta(1,1) prior is equivalent to a Uniform(0,1) prior and could easily be changed to the Jeffreys's beta(0.5,0.5) prior or anything you desire.