Migrate SQL server data to new schema
Migrating the Data is a Job in itself
Keeping Data in sync is a job is yet another job. 
Creating a new front end is yet another job. 

there's lots of ways to tackle the problem. 

there's a lot that can go wrong with data migrations, so if you've never done one before unless the business is yelling for it I'd move very slowly and carefully. 

1st I'd replicate the Schema and just do a simple import into sql. 

From there build out what you want your new schema to look like and build your scripts for moving data around into your new schema, make sure you're logging every single insert update delete etc, it's easiest if you log the command that you ran to move it, and the Command to undo the operation in a central table. 

then build your new front end against that and test it thoroughly. 

then test your scripts for exporting from access and importing into your new schema, then you can automate the Export import process until it runs perfectly every time I'd probably do at least 100 or so executions to test the process before the move. 

I wouldn't worry about keeping it in sync until you've got buy in from the business with your new process and application then I'd just set a time for the Switch. 

If you have to have 2 simultaneous systems In place you've got a whole different set of problems. 

Which one becomes the source of truth? do you need to be able to export everything back out to access at any given time. 

Just about any change in scope will add additional time to the Project. 

Hit me up if you have any other questions. I've done a couple around 50 or so of these small scale data migrations from flat files to sql, from access to sql, from sql to access, oracle to MongoDB, from SQL to Mongodb, mysql to sql, mongodb to SQL and they take a lot longer than you'd think if you want to do it right.