Transformer neural networks are increasingly replacing prior architectures in a wide range of applications in different data modalities. The increasing size and computational demands of fine-tuning large pre-trained transformer neural networks pose significant challenges for the widespread adoption of these models for applications that demand on-edge computing. To tackle this challenge, continual learning (CL) emerges as a solution by facilitating the transfer of knowledge across tasks that arrive sequentially for an autonomously learning agent. However, current CL methods mainly focus on learning tasks that are exclusively vision-based or language-based. We propose a transformer-based CL framework focusing on learning tasks that involve both vision and language, known as Vision-and-Language (VaL) tasks. Due to the success of transformers in other modalities, our architecture has the potential to be used in multimodal learning settings. In our framework, we benefit from introducing extra parameters to a base transformer to specialize the network for each task. As a result, we enable dynamic model expansion to learn several tasks in a sequence. We also use knowledge distillation to benefit from relevant past experiences to learn the current task more efficiently. Our proposed method, Task Attentive Multimodal Continual Learning (TAM-CL), allows for the exchange of information between tasks while mitigating the problem of catastrophic forgetting. Notably, our approach is scalable, incurring minimal memory and time overhead. TAM-CL achieves state-of-the-art (SOTA) performance on challenging multimodal tasks