In theory, they could. But IMHO it would not be wise to overestimate the EU’s competence when it comes to purchasing software projects and writing a sane requirement specification.
As a citizen within EU I would rather see them donate money to the Signal Technology Foundation instead. OTOH that could be controversial, so perhaps it would be better if citizens and companies would donate directly instead.
Perhaps they could establish a satellite office in the EU with developers that are officially [0] hired by the EU but work for Signal? That way there could be a purely EU owned fork of Signal.
[0] Signal conducts the hiring and interviews, EU pays the salary
The EU's budget is (was) just 4% of the US' budget. That's still 148 billion Euro, but the comparison might show why the EU isn't quite as keen to "invest in <x>" as people sometimes seem to think. (Although one might consider these suggestions a reflection of the EU's connection with dreaming big in people's minds)
The EU's structure is also a bit different than that of nation states, with cabinet members and parliament rather limited in their ability to make spending decisions. Anything serious needs approval by the council, i. e. the heads of state.
And just now is probably a bad time for new expenditures, as Brexit will severely cut into revenue.
