My garden is boggy with bad drainage, would this be a problem?

I want to install artificial grass in it so I can finally have a beautiful garden.

No, it won't be a problem. Here is why:
First, you need to make sure to follow the proper installation instructions on this link:
https://megagrass.com/pages/installation
The very first step you need to do is to excavate about 4 inches of the your boggy soil, then you need to use drain rock along with decomposed granite (or quarter minus or similar materials) to build your base.
Once your base is built and compacted properly, you create a new drainage system for your artificial grass.
So instead of draining through your boggy surface, water will drain through the newly built drainage system for your artificial turf.
Therefore, your current boggy soil won't present any drainage concerns any more.

Please always feel free to reach us for more expert's opinions by fill out this online form:
https://megagrass.com/pages/free-consultation

I've seen some people who have worse garden conditions. As long as you have a great installer, I'm sure they'll be able to fix that for you.