There was a sort of Roman Postal service called the Cursus Publicus that was set up by Augustus as an emulation of the famous Persian service. The Cursus Publicus had stations strategically placed along Roman roads where you could dispatch riders to ride until their horse was spent, and they could pick up a new horse/transfer their message to a new rider. The system was also used for government officials to travel throughout the Empire.
The problem with this as an answer to your question, however, is that the Cursus Publicus was not open to average people; you had to have a special diploma from the Emperor to use it (although there is evidence in the Theodosian Law Code, which includes a law cracking down on the usage of the Cursus, that the system was abused and there were forged diplomas). Individuals generally had to rely on friends or slaves to carry their letters; this is how, for example, Cicero and his BFF Atticus corresponded (we have a large number of their letters).
Poorer individuals like soldiers, who may not have had the wealth to have a slave carry a letter and/or package would have had to rely on soldiers returning to their areas to bring the letter and another one to return a response. This would have been an incredibly slow process and lots of the Roman epistolary tradition includes admonishments and complaints of long delays between letters. So it could be done, but slowly, and things were often lost along the way (other complaints of lost letters are also fairly common).
This won't completely answer your question because 1. I can only really speak to soldiers stationed in Britain, and 2. Your specific instance of soldiers asking for things from their families is outside of examples that I know of. From the letters I'm familiar with, primarily those from Vindolanda, what we have records of are soldiers buying/sending items to their families, or soldiers higher along the chain of command monitoring or ordering goods, both within a military capacity and as a head of a household. As far as the movement of these goods mentioned in the letters, you have a few options:
Merchants selling goods could also be responsible for the movement of those goods after purchase, as is the case for certain merchants contracted by the military.
We have records of soldiers being dispatched specifically for the purpose of transporting goods or purchasing others.
It is also worth considering that soldiers at all levels within forts participated in slavery. While the role of a common legionnaire's slave may have been more limited, we know from correspondence at Vindolanda that slaves had crucial roles in maintaining households, which meant overseeing and ordering goods, while the soldiers (more accurately officers, like the legati) attended military duties. Lower ranking slaves may have then been responsible for actually travelling and acquiring the ordered goods.
And as seems to be the case from Vindolanda, the families of the Roman military often went with the soldiers to their post, living in the large domestic settlements that popped up outside forts. So while correspondence was still made between soldiers and their families, the requested transactions were happening locally, rather than over the distances between Rome and the frontier.
I hope that helps, and if you can find a specific example of soldiers requesting things from their families, I'll do some research and adjust my answer.
All references to operations at Vindolanda and its tablets are taken from: "Life and Letters on the Roman Frontier: Vindolanda and its People" by Alan Bowman 2003