The Roman army in the Republic was a seasonal position with soldiers signing up for a campaign in the summer, then going home to work the fields. Some of the soldiers wives would join them in the baggage train, however they were not permitted to stay in the camp. It wasn't until Augustus that one could pick soldier as a full time career.

If one was married before joining the army, the marriage was nullified. However, like Theemuts pointed out, many soldiers did form long term relationships with local women, many having kids with them. Men of Equestrian and senatorial rank were able to remain married to bring their wives into the camp. Legionaries got special exemptions while writing their will, and could leave their inheritance to a non-Roman.

Source The Roman Army. Southern, Pat

I also know that Roman soldiers were given leave however I need to find a source for it.

This has grown quite source-heavy, which I originally didn't expect, but this is an interesting legal topic and so I rely quite a bit on legal texts and original sources. I will also only discuss desertion in the context of being granted leave and staying away for too long, to make things a bit more straightforward. If anything is unclear, please ask away.

Roman legionaries were technically not allowed to have legally acknowledged wives until Septimius Severus in the late 2nd century, but this was something more important in the legal sphere (f.e. in hereditary law, if a soldier wanted to leave stuff to his non-legal non-citizen wife) than in actual personal relations, f.e. with women living near camp, of which Roman soldiers had plenty.

Recruitment was probably close to 50/50 between locally recruited soldiers and those that came from further abroad for much of the Empire; from Tiberius and Nero on the reliance on regional and even local recruitment increased, more so for the legions than many specialized arms like archers (who were predominantly recruited from the Near East) or cavalry (Gauls, Germans or Hispanics were preferred here). These are only general numbers and trends, and it is hard to get a good picture of these practices in detail from the sources that are available to us; but the general trend in the legions is towards recruitment from among the society that the legion was stationed in, or even from the castrum and its surrounding village itself. So while many wouldn't have required much time for leave to visit their family, for others this might involve a quite long trip, for some through the whole empire (one of the places where the aforementioned Syrian archers were stationed, f.e. included Britain and Germany).

Naturally, though, as in any army, leave is a common problem for soldiers and leaders alike: the former want what the latter are reluctant to grant. But in general, roman leadership recognized the interest of the soldiers in being granted vacation, for various reasons, such as vacation with the family at home, or private interests. It was a necessary evil, so to speak, since most of our sources mention the negative aspects of granting soldiers leave, so it was probably handed out sparingly.

In Vegetius Epitoma rei militaris, a treatise on military science, the practice of granting leave is seen critical:

When anyone receives leave of absence (commeatus), and for how many days, it is noted down in lists. For in antiquity (referring here to the early empire) it was difficult to be given unless for very good approved reasons. It seemed incongruous that a soldier of the Emperor, maintained in uniform and pay and rations at public expense, should have time to serve private interests [II.19].

Commenting on the tendency of units stationed for a long time in the same place in peacetime towards decadence and mutiny:

A compound treatment is usually applied to this wound. While they are still separate and in their base, (soldiers) should be held to every article of discipline by the strictest severity [...], they should have opportunity for no leave of absence [III.4]

Aemilius Macer, a roman legal scholar from the time of Severus comments on early jurisprudence, in reference to a law of Augustus regarding discipline:

Paternus also wrote that a conscientous leader of soldiers should grant leave very sparingly [...] [Dig. 49.16.12, 1]

Accordingly, Suetonius portrays Augustus as a very strict disciplinarian:

Disciplinam severissime rexit. ne legatorum quidem cuiquam, nisi gravate hibernisque demum mensibus, permisit uxorem intervisere.

"In discipline, he ruled most severe. He allowed the legates to visit their wives only with great reluctance, and even then only in the winter months (i.e. out of campaigning season)."

Galba, too, is said to have been very strict with regards to this practice:

with equal severity he forbid [the soldiers] to ask for leave. [Suet. Galba VI.2]

Some of the problems of granting too much leave are illustrated by Tacitus, here during the parthian invasion of Syria and Armenia:

The bridge was now complete, and the hills in front were occupied, [...] with a speed and a display of strength which induced the Parthians to drop their preparations for invading Syria and to stake their whole hopes upon Armenia; where Paetus, unconscious of the coming danger, was keeping the fifth legion stationed far away in Pontus, and had weakened the rest (the fourth and twelfth legions) through unrestricted grants of leave, until he heard that Vologeses was coming with a large and threatening army. [Tac. Ann. XV.9, 2]

L. Caesennius Paetus, as imperial legate, was caught with his pants down - the Parthians invaded, and his subordinates had granted too much leave to his troops! He would suffer a crushing defeat with the 4th and 12th legions against Vologaeses in the battle of Rhandeia. Though Tacitus might embellish the impact of leave here to explain Roman defeat in battle, this makes two things clear: One, the roman leadership viewed the granting of leave as something to be avoided, as detrimental to discipline and good troop leadership, and two, that it was still granted and sometimes in unhealthy amounts. And in essence commeatus, or vacation (originally meaning something like 'unhindered coming and going') was something that was legally available to all ranks and all kind of troops, legionary or auxiliary, though there was no legal right to a certain amount of vacation per year. Also, our sources suggest, in light of the above, that this was not something like modern vacation for the purpose of recreation, but something that was only granted for valid reasons.

Luckily, a few contemporary documents elucidating aspects of this topics have survived (mainly Papyri from egypt, where the conditions for survival were excellent). On of the reasons documented there was looking after personal possessions somewhere else (others might be medical reasons, or reasons related to politcal elections):

to the decurion Teres, greetings. [?]stermus, strategos of the Coptite nome (the administrative sub-divisions of Egypt, of which a strategos was in charge of) has asked me to grand you 30 days' leave so that you may visit your possessions which you have in the Arsinoite nome. I have written not only to him but also to the centurion Petronius Fidus that I grant this [P.Wisc. 2.70]

This tells us also something about what a highly bureaucratic affair this was, and much writing and lists were involved in the process (see above, where Vegetius tells us that this was written down in lists). Such written letters probably also served as token of proof for the individual soldier that he had been granted leave. One of these lists that suvived as Papyrus from Egypt [O.Flor.1] even specifies that a soldier was to be granted 10 days of vacation (δέκα ι ἡμέρας κο- μιᾶτεν = commeatus) plus two more days for travelling (ἔχεις δύο ἡμέρας κο- μιᾶτεν καὶ σοὶ ἀπαγαθαι). So not only was the time (quite) limited, travelling time was taken into account as well. What's interesting, too, is that the name of the soldier, Ammonas, was added by a second hand, so it might have even been a blanco formular!

Another soldier, Iulius Apollinarius wrote a letter to his father Iulius Sabinus in March 107, telling him about his life in the army and his dreams of advancement, but also expressing his hope for some leave to visit his family:

I shall take pains, as soon as the prefect (of the province?) begins to grant furloughs, to come to you immediately. [P.Mich. 8.466]

So, yes, visiting family seems indeed to have been a valid reason, though distance and time might have been a limiting factor here. Often soldiers would, despite the ban on offically sancioned marriage, find de facto wives and families in the villages that sprang up around the legionary camps everywhere, the so-called canabae.

Now, we have seen that soldiers could, grudgingly, be granted vacation, and even for such mundane reasons as visiting their families. But, today as back then, soldiers away from base are known to 'get lost' or be late in return. Obviously this is a serious violation of military discipline (whose values we have seen being extolled above), and that must be punished lest the unit's discipline as a whole suffer.

The later Roman Empire (second century onward) saw legionaries and auxiliaries spend increasingly more time in their fortified encampments and forts. Over time, the forts turned into mini Roman colonies and resembled a walled city, more than a temporary Roman encampment. Local people along with the families of Roman soldiers would often live in or outside the walls. Thus, many of the soldiers had families around their military stations and would not go on leave to visit them.

This is in contrast to the early and late Republic era which saw campaigns and enlistments revolve around seasons, meaning legionaries would return around harvest time to their families each year.

Follow up question, could you ONLY enlist for 25 years?

Followup question: What was the age of enlistment?

Was under the assumption that during the Republican era, only landowners could join the legions.