If your license has expired, you can obtain a new one through the following procedure.
Literal
License [possessive-の] validity-period [subject-が] expired [case-場合は], following method [with-で] new license [possessive-の] issuance [object-を] receive [potential-ことができる].
Official/legal language. 有効期間が切れてしまった ('the validity period has expired') uses ~てしまう for the unfortunate completion. 場合は ('in the case that') is the standard conditional for official instructions. 交付を受ける ('to receive issuance') is bureaucratic vocabulary — the passive-like phrasing where one 'receives' an official action. ~ことができる is the formal potential.