First, more information on the general subject of shin splints can be found here, and as such I will not repeat it below.
Yes and no.
The military does not ‘allow’ shin splints. Shin splints is typically a symptom of the formal/informal training that an individual follows. Simplistically, if your shin splints is caused by training then adaptation of your training and/or rest should be enough to ‘solve the problem’ and would not be a barrier to joining the military.
However, it must be noted that there are also a number of other causes and if symptoms persist then seek advice from a medical professional. Depending on the diagnosis (what the problem is) and prognosis (what is the likely outcome) this may or may not be a barrier to joining the military.
Recruitment & Selection Process
As part of the military recruitment and selection process a medical professional (in the first instance typically your own GP) will take a detailed history from you. Although there is not a specific question about shin splints as part of this process, the medical professional may ask qualifying questions based on answers given by you. If shin splints are diagnosed at this stage your GP is likely to suggest rest/change of training followed by a review after a period of time.
At the Selection Centre a thorough medical examination will be undertaken, as well as a fitness test. During this a medical professional will conduct a thorough review of your medical notes and the results of the medical examination in order to make an informed judgement on your medical suitability to undertake military training and engage in a productive military career. If shin splints are diagnosed at this stage the medical professional is likely to suggest that you re-attend the Selection Centre after a period of rest and/or change of training; probably with a review by your own GP.
Phase 1 Initial Military Training
Once ‘in the system’ you will be cared for by the medical centre attached to your training establishment if you are unfortunate enough to suffer an injury. If shin splints are diagnosed at this stage you will likely go on ‘light duties’ meaning no PT (physical training) and be referred to the physiotherapist who will develop an exercise rehabilitation programme of strengthening exercises. Depending on the duration of this, you may be ‘back trooped’ (meaning transferred to another troop at the same stage of training you left) as the physical training on the course will have progressed beyond your previous state.