Sita and Rama waiting for kids for a long time can be reasoned in four ways.
It's completely their wish.
Having kids and when to have kids is upto the wish of the couple. Both sita and Rama know their dharma. Rama is hailed as embodiment of Dharma in Valmiki Ramayana. Moreover, there is no evidence of a person or a situation that forces them to have kids very early. So we can leave that situation upto them.
Marriage of Rama and Sita happened at a very early age. Lord Rama was 13 years old. Sita was 6 years when marriage happened. Later they spent 12 years happily in Ayodhya.
मम भर्ता महातेजा वयसा पंच विंशकः || अष्टा दश हि वर्षाणि मम जन्मनि गण्यते | (Aranya Kanda 47 Sarga 11 sloka)
"My great-resplendent husband was of twenty-five years of age at that time, and to me eighteen years are reckoned up from my birth"
समा द्वादश तत्र अहम् राघवस्य निवेशने || ५-३३-१७ भुन्जाना मानुषान् भोगान् सर्व काम समृद्धिनी |
I stayed in Rama's house there for twelve years, enjoying the worldly pleasures belonging to human kind and fulfilling all my desires."
ततः त्रयोदशे वर्षे राज्येन इक्ष्वाकु नन्दनम् || ५-३३-१८
अभिषेचयितुम् राजा स उपाध्यायः प्रचक्रमे |
Thereafter, in the thirteenth year, King Dasaratha along with his preceptors started to perform anointment of the kingdom to Rama, a celebrity of Ikshvaku dynasty
See Karmanya Nanda's answer to see exact age and calculation.
Rama went for an exile, not for a tour
They were in teenage when they spent their happy time in the city of Ayodhya. A girl in her early teen cannot give birth to healthy child. So they had to wait for some more time to have children.
When they reached an age of an adult, Rama and Sita went for an exile. Rama was ordered to lead a life of an ascetic.
He says following words to his mother Kausalya before leaving to the forest.
मन्युर् न च त्वया कार्यो देवि ब्रूहि तव अग्रतः |
यास्यामि भव सुप्रीता वनम् चीर जटा धरः || २-१९-४
"Oh, queen! you need not be indignant. I am telling before you that I shall go to the forest, wearing rags and braided hair. Become delighted well."
Even though Rama objected Sita, Sita accompanied Rama to the forest. They lived in a hermitage. They lived a life of an ascetic with braided hair, wearing rags.
In that situation, how are they supposed to think of having children? It is a bhoga (pleasure). They spent time in the forest but didn't think of progeny during their forest.
Extra risk of having a child in a forest.
Let us consider they thought of having progeny, then what would be the situation of pregnant woman Sita? Who shall take care of her in the forest? There are only two men i.e., Rama and Lakshmana. They can take help of some rishis but that would be against their vow of living an independent life in the forest. Remember the boon is living in the forest but not as another's guest.
Forest is not an ideal place for humans. Forests of those days not only had wild animals, there were rakshasas too. If there were children, they should not only take care of themselves, a kid is an extra responsibility. For warriors like Rama and Lakshmana, it's not difficult but it's a risk to live in a forest with infants.
Rama is a prince (because he was not yet coronated as a king). His children would take birth in a palace if he waits for a few years. He can very well wait for some years thinking of welfare of his children and wife.
Lakshmana also went to exile along with his brother just for service. For a person like Rama, it doesn't sound nice to think of pleasures when younger brother is away from wife.
These are the reasons why Rama and Sita waited for long to have children.