I actually went and plugged in the numbers for coagulation disorders into a frequentist likelihood ratio test for a difference in proportions, and got p = 0.23.

Feel free to check my work (in R):

coag_vax <- 16
p_vax <- 8.5 / 10000
num_vax <- round(coag_vax / p_vax)
cat("#(vaccine) = ", num_vax, "\n", sep = "")
cat("#(coagulation disorder | vaccine) = ", coag_vax, "\n", sep = "")
cat("P(coagulation disorder | vaccine) = ", p_vax, "\n", sep = "")
cat("\n")

coag_plac <- 10
p_plac <- 5.3 / 10000
num_plac <- round(coag_plac / p_plac)
cat("#(placebo) = ", num_plac, "\n", sep = "")
cat("#(coagulation disorder | placebo) = ", coag_plac, "\n", sep = "")
cat("P(coagulation disorder | placebo) = ", p_plac, "\n", sep = "")
cat("\n")

coag_same <- coag_vax + coag_plac
num_same <- num_vax + num_plac
p_same <- coag_same / num_same
cat("#(null) = ", num_same, "\n", sep = "")
cat("#(coagulation disorder | null) = ", coag_same, "\n", sep = "")
cat("P(coagulation disorder | null) = ", p_same, "\n", sep = "")
cat("\n")

## Frequentist likelihood ratio test that the proportions are identical

# https://stats.stackexchange.com/a/373448/36229
# http://people.musc.edu/~bandyopd/bmtry711.11/lecture_02.pdf
l_same <- dbinom(coag_vax, num_vax, p_same) * dbinom(coag_plac, num_plac, p_same)
l_diff <- dbinom(coag_vax, num_vax, p_vax) * dbinom(coag_plac, num_plac, p_plac)
t <- -2.0 * log(l_same / l_diff)
p <- 1 - pchisq(t, 1)
cat("LR test statistic = ", t, "\n", sep = "")
cat("LR test p-value = ", p, "\n", sep = "")

Output:

#(vaccine) = 18824
#(coagulation disorder | vaccine) = 16
P(coagulation disorder | vaccine) = 0.00085

#(placebo) = 18868
#(coagulation disorder | placebo) = 10
P(coagulation disorder | placebo) = 0.00053

#(null) = 37692
#(coagulation disorder | null) = 26
P(coagulation disorder | null) = 0.0006898015

LR test statistic = 1.412184
LR test p-value = 0.2346942

And for a 95% CI around the difference in proportions, I got [0, 8.5]:

# https://pubmed.ncbi.nlm.nih.gov/25163425/
# var_vax <- num_vax * p_vax * (1 - p_vax)
# var_plac <- num_plac * p_plac * (1 - p_plac)
var_pooled <- (p_vax * (1 - p_vax) / num_vax) + (p_plac * (1 - p_plac) / num_plac)
std_pooled <- sqrt(var_pooled)
ci_lo <- (p_vax - p_plac) - qnorm(0.975) * std_pooled
ci_hi <- (p_vax - p_plac) + qnorm(0.975) * std_pooled
cat("95% CI (Gaussian approximation) = [ ", max(ci_lo, 0.0) * 10000, " in 10k , ", min(ci_hi, 1.0) * 10000, " in 10k ]\n", sep = "")

Output:

95% CI (Gaussian approximation) = [ 0 in 10k , 8.502481 in 10k ]

I also ran it through a completely-un-tuned Bayesian model and got a similarly inconclusive result:

library(brms)

coag_data <- data.frame(
category = c("vaccine", "placebo"),
n_sample = c(num_vax, num_plac),
n_coag = c(coag_vax, coag_plac))

fmla <- n_coag | trials(n_sample) ~ category

prior <- get_prior(fmla, coag_data, family = binomial)
cat("Prior:\n")
print(prior)

model <- brm(
fmla,
data = coag_data,
family = binomial(link = "logit"),
prior = prior,
iter = 1000)
cat("Model:\n")
print(model)

Output (truncated):

Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept -7.58 0.32 -8.23 -6.98 1.00 1000 1278
categoryvaccine 0.49 0.41 -0.27 1.32 1.00 1319 1239

Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

Conclusion: It's suggestive that the vaccine is more dangerous than the placebo but not at all a strong confirmation.

I did skim back over the paper and it looks like they categorized "serious" in such a way that maybe hospitalization is not a terrible baseline for comparison. But it's still pretty questionable that the incidence of "serious" coagulation disorders in the placebo group is higher than the incidence of hospitalization in the placebo group, and it makes me think that they're making an invalid comparison there. So this is an inconclusive result, presented in what is at best a hand-wavey framework.