Does SEM test the significance of a mediator? Does structural equation modelling inherently test the significance of the indirect effect (i.e. the mediator)? For example, other methods of mediation such as regression use bootstrapping to test the significance of the indirect effect. I do not know much about SEM so I'm wondering whether this is tested inherently in the model or whether there needs to be an add-on (e.g. bootstrapping). 
 A: A structural equation model can be specified to test the indirect effect, but it does not "inherently" do it. Most modern methods of determining if there is an indirect effect multiply the regression coefficients for the $a$ and $b$ paths, and then use bootstrapping to get a 95% confidence interval around it.
What you can do is label the $a$ and $b$ paths, define the indirect effect as $a \times b$, and then get bootstrap CIs around each parameter.
If you are working in R, I have a .pdf, .R script, and sample data that shows you how to do moderation, mediation, and moderated mediation in R.
To get a mediation effect, here is the relevant code:
model4 <- "med ~ a*iv
           dv ~ cp*iv + b*med
           mediate := a*b"

The mediator is predicted by the IV, which is named $a$. Then the DV is predicted by both the IV and mediator, where the mediator's influence is named $b$. Then the mediation effect, mediate is $a \times b$.
model4.fit <- sem(model=model4, data=data, se="boot", bootstrap=5000)
model4parameters <- parameterEstimates(model4.fit, boot.ci.type="bca.simple")
model4parameters

That first line will fit the model using bootstrap confidence intervals, and then the second and third lines will give you the parameter estimates. You would just look for the "mediate" effect there.
Note that iv, dv, and med there are all indicator variables. You can define them earlier in the code as latent variables if you would like by saying, perhaps, that iv is made up of four different items:
iv =~ iv1 + iv2 + iv3 + iv4

And similar code could be done for the other variables to make it a true structural equation model (i.e., it has both measurement and structural parts; the code I presented above is just a path model with the structural parts).
