There will be no tax implications of doing a ROTH IRA transfer from one custodian (Vanguard) to another (Fidelity). You will simply go to the institution that you are transferring to and fill out the appropriate paperwork or online form. Fidelity (in your case) will initiate the transfer. Depending on how your ROTH IRA is invested, you may want to sell shares of the funds you are invested in if Fidelity charges you a transaction fee for the particular securities.

The amount you transfer to the new custodian will not contribute toward the $5500 annual amount for 2018 as long as you didn't contribute those monies this year.

There will be no tax implications of doing a ROTH IRA transfer from one custodian (Vanguard) to another (Fidelity). You will simply go to the institution that you are transferring to and fill out the appropriate paperwork or online form. Fidelity (in your case) will initiate the transfer. Depending on how your ROTH IRA is invested, you may want to sell shares of the funds you are invested in if Fidelity charges you a transaction fee for the particular securities.

Transferring money to a new custodian does not impact nor does it count toward the annual $5500 contribution limit.