Couldn't they rent some fields/stadia from the City of Rome to do it to make up for the lack of space within the Vatican walls? The local soccer league in my city rents fields from the local Boys' and Girls' Club for the youth soccer season every year (granted, it's in the fall, but it's the idea that counts).
They would
have to, which is why any Vatican team would have to be inside the Italian football system, and not independent like sanmarinese teams.
A skeleton TL: One of the founding friars of Fortitudo-Pro Roma SGS somehow manages to get himself elected Pope in the 1920's. Fortitudo wasn't a bad team -they won the Southern Italy championship in 1922, so maybe a few more successes during the 20's give the team a popularity boost during the decade, specially in its home area around the Vatican. When Mussolini forces all roman teams to merge into AS Roma in 1927, the Pope intervenes to prevent his old team from disappearing.
The 1927-1928 Italian Championship is played in a series of regional sub-leagues to determine which teams will make it to the first national Serie A in the 1928-29. Fortitudo has a rocky campaign, as referees suffer political pressures to favour Roma and Lazio and is selected to the Serie B. However, during the 30's, Papal support allows them to develop and be promoted to the Serie A in the 1937 season. While the Pope's majesty prevents him from attending matches, it is a well known fact in Rome that he is a rabid Fortitudo supporter, he religiously follows their matches by radio and gives them financial and moral support. When the Pope dies in 1939, the team is consolidated as the third Roman team -even if it keeps struggling in the lower positions of the table, never getting near the championship-, and it finds an unlikely fanbase in Rome's antifascists, who can cheer for a local team that is untainted with fascist associations.
This continues after the war, with the team being unnoficially known as the Vatican team. During the 40's and 50's Fortitudo is Rome's minor team, getting demoted to Serie B a couple of times, but always being promoted back to Serie A, and making it to the Italian Cup final in 1949 and 1953.
However, in 1961 the team is in financial dire straits. John XXIII (himself a Venice supporter) sees this as a golden opportunity in his
aggiornamento policy and does not hesitate in buying the club, renaming it as Fortitude Vaticano FC and changing the club's colours from red-blue to white-yellow.
However, Pope John's most important legacy will be the creation of the Vatican Football Academy and the Pope's definition of Fortitude Vaticano's moral and social role: the team's role is to reach to the youth by exemplifying catholic virtues, both on and off the field. Obviously, Fortitude Vaticano does not engage in simply trading players, but will feed from the Vatican Football Academy, which trains poor children from Rome's suburbs and gives them a second opportunity, with those who don't make it to the team being prepared for ecclesiastic life.(This system is similar to Athletic de Bilbao's in Spain, who has played in Primera Division for 83 straight seasons and won two spanish leagues using only Basque players or players breeded in Athletic's academy)
Many mock the Pope for his bold gesture, but during the 60's and 70's Fortitude keeps growing: in 1963 they finish above arch-rivals Roma and Lazio in Serie A for the first time; and finally, in 1966 they win the Italian Cup after beating Fiorentina in the final. Not only is this their first title since the 1920's, it also gives them a spot in the European Cup Winners' Cup for the 1966-1967 season.
The first round draws them against soviet champion Spartak Moscow. The world awaits the expected duel between the vatican and soviet team, which never happens since Spartak does not show up for the first leg in Rome for political pressure. This is a major moral victory for Fortitude and catholic propaganda around the world, at least until Rangers FC beats them in the next round...
In 1968, the Italian team that wins the European Championship beating Yugoslavia has two Fortitude players in its ranks, both product of the Vatican Academy -one in his way to be an ordained priest. Fortitude continues to grow during the 70's, getting a couple of UEFA cup spots thanks to good shows in Serie A. Their 1978 Round of 16 eliminatory against Legia Warsaw (the Polish Army club!) is credited to be one of the factors leading to the Polish awakening due to the crackdown after the home crowd booed Legia and supported Fortitude during the match leg played in Warsaw. In 1980 John Paul II becomes the first Pope to attend a match at Fortitude's home field Stadio Flaminio, during an otherwise unremarkable league match against Napoli. From then on, Papal presence during matches, while not granted, is at least not uncommon.
In 1982, three Fortidude players play in the Italian National Football team that wins the World Cup in Spain. Three seasons later, after an outstanding season and a last minute shocking win against AC Milan in San Siro, Fortitude Vaticano wins its first Serie A title.
All of Rome celebrates, and while the team is not allowed to present the
scudetto from St Peter's balcony, the Pope allows the crowd to celebrate in St Peter's square. Fortitude plays its first European Club Championship in the 1985-1986 season, being beaten in the quarter finals by Real Madrid, after eliminating CSKA Moscow in the previous round, in a very politically charged match in Moscow.
In 2011, Fortitude Vaticano is one of Italy's most famous teams, having won two more
scudetti in 2001 and 2007, and becoming a Champions League regular after the competition was modified in 1993, their best showing a semifinal defeat by penalties against Manchester United in 2004. During the 1980's, the Vatican Academy opened new sucursals in Africa and South America, attracting some local talent. Fortitude is famous for its traditionally attacking football -a rarity in the italian league- and its very wide fanbase, since most devout catholics around the world consider it their second team along with their local team.