DNIPRO, Ukraine—One Ukrainian paid almost $10,000 to flee the draft. Another has ignored five military summonses. A third avoids public spaces, fearing a military official will pounce and issue a call-up. After a year of war, Ukraine is facing increasing challenges in raising the troops it needs to resist Russian forces and eject them from its territory. When Russia invaded in February last year, thousands of volunteers lined up outside military recruitment centers. With many of them now dead or injured, Ukrainian authorities are scrambling to recruit replacements, often drafting those who have neither the desire nor the training to...