The C standard library’s time.h is includable.
#include "Arduino.h"
#include <time.h>
void setup() {
time_t currentTime = time(NULL);
}
void loop() { }
compiles – the question is what exact functionality do you expect from it. The Atmega328P has no real-time-clock / RTC peripheral to hold the time. There are workarounds with timers et cetera, but these are not given by the standard C library.
For vector.h: Do you really want to use the Standard Template Library, and this datastructure, which allocates its elements on the heap? On such a memory constrained device like a ATMega328P? Not a good I think. AVR-GCC doesn’t have <vector.h> by default – you have to use libraries which fill the gap there, as described in Arduino c++11 or c++14 support in platform IO - #3 by maxgerhardt.