I found an answer to my question, and I share it below in case it might be helpful for others.
I am still not sure if one can obtain a 95% RMSEA CI directly in lavaan, but the MBESS package can provide that. You just need to install the package and have the following information: RMSEA (from the model fit output), sample size, and degrees of freedom.
Here is the code I used:
#Change CI of RMSEA ----
## Get values needed to change RMSEA CI
fitmeasures(fit, c("rmsea", "rmsea.ci.lower", "rmsea.ci.upper")) # Get RMSEA and its CI
lavInspect(fit, what = "nobs") # Get N used in model
fitmeasures(fit "df") # Get the dof in model
### Install MBESS package to be able to change RMSEA CI
install.packages("MBESS")
library(MBESS)
###
MBESS::ci.rmsea(rmsea=.052, df=328, N=379, conf.level=.90) #Gives 90% CI for RMSEA
MBESS::ci.rmsea(rmsea=.052, df=328, N=379, conf.level=.95) #Gives 95% CI for RMSEA