Vector::push_back is there for STL compatibility, but you don't really need it here. You can simply use Add method, as there is already a constructor (String(const std::string& s)) for conversion.
E.g:
#include <Core/Core.h>
#include <vector>
using namespace Upp;
CONSOLE_APP_MAIN
{
StdLogSetup(LOG_COUT);
std::vector<std::string> sWords;
Vector<String> uWords;
for(int i = 0; i < 5; i++)
sWords.push_back(std::to_string(i));
for(int i = 0; i < sWords.size(); i++)
uWords.Add(sWords[i]);
for(const auto& s : uWords)
LOG(s);
}